Matrices and Determinants
Contents
Describe and classify Matrices 1
Differentiate between Singular and Nonsingular Matrices 4
Describe Algebraic Operations on Matrices 6
Describe Elementary operations or Transformation of a Matrix 9
Calculate the Inverse using Elementary operations 9
Describe and Calculate Transpose of a Matrix 9
Describe Symmetric and SkewSymmetric Matrices 9
Describe Orthogonal Matrix and its properties 9
Describe Complex conjugate of a Matrix 9
Describe Hermitian and Skew Hermitian of a Matrix and its properties 9
Describe Unitary Matrix with example 9
Describe Determinant and its properties 9
Describe and apply Laplace’s Method of Expansion of Determinant 9
Describe Adjoint of a Matrix and its Properties 9
Describe Adjoint of a Determinant 9
Apply Adjoint to find Inverse of a Matrix 9
Describe Linear Simultaneous Equations 9
Describe the steps to calculate the solutions of Linear Simultaneous Equations using Cramer’s rule 9
Describe the systems of Homogeneous and Nonhomogeneous equations 9
Describe Eigenvalues and Eigenvectors and their properties 9
Describe Characteristic equation with example 9
Describe CayleyHamilton theorem and its application 9
Describe Vector Space with examples 9
Describe Subspace of Vector Space with examples 9
Describe Linear dependence and Independence of vectors 9
Describe Linear Transformation 9
Verify the given Mappings on Linear or Nonlinear Transformation 9
Describe Matrix representation of a Linear Operation 9
Describe Kernel and Image of a Linear Mapping 9
Describe Rank and Nullity of a Linear Mapping and discuss the RankNullity Theorem 9
Calculate Kernel and Image of a Linear Transformation 9
Calculate the Rank of a Matrix using Echelon Form 9
Calculate the Rank of a Matrix using Normal Form 9
Describe and classify Matrices
This learning outcome requires you to be able to describe and classify matrices. Matrices are arrays of numbers arranged in rows and columns, which are used in a variety of fields such as mathematics, computer science, and engineering. Matrices can be used to represent data, perform mathematical operations, and solve systems of equations.
To achieve this learning outcome, you should be able to:
 Define what a matrix is
 Describe the different types of matrices
 Classify matrices based on their properties
 Perform basic matrix operations
 Solve systems of equations using matrices
Let’s discuss each of these in more detail:
Definition of a matrix:
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are denoted by capital letters, such as A, B, or C. The size of a matrix is denoted by its dimensions, which specify the number of rows and columns in the matrix. For example, a matrix with m rows and n columns is denoted as an “m x n” matrix.
Example:
A =
 2 5 1 
 3 6 2 
In this example, A is a 2 x 3 matrix with 2 rows and 3 columns.
Types of matrices:
There are several types of matrices, including:
 Square matrix: A matrix with an equal number of rows and columns is called a square matrix. For example, a 3 x 3 matrix is a square matrix.
 Diagonal matrix: A square matrix where all the elements outside the diagonal are zero is called a diagonal matrix.
 Identity matrix: A diagonal matrix where all the diagonal elements are 1 is called an identity matrix. It is denoted by I.
 Zero matrix: A matrix where all the elements are zero is called a zero matrix. It is denoted by 0.
 Row matrix: A matrix with a single row is called a row matrix.
 Column matrix: A matrix with a single column is called a column matrix.
Example:
A =
 2 5 
 3 6 
In this example, A is a 2 x 2 square matrix.
B =
 1 0 0 
 0 1 0 
 0 0 1 
In this example, B is a 3 x 3 identity matrix.
Classification of matrices:
Matrices can also be classified based on their properties. Some common classifications include:
 Symmetric matrix: A square matrix where the element in the ith row and jth column is equal to the element in the jth row and ith column is called a symmetric matrix.
 Skewsymmetric matrix: A square matrix where the element in the ith row and jth column is equal to the negative of the element in the jth row and ith column is called a skewsymmetric matrix.
 Upper triangular matrix: A square matrix where all the elements below the diagonal are zero is called an upper triangular matrix.
 Lower triangular matrix: A square matrix where all the elements above the diagonal are zero is called a lower triangular matrix.
Example:
A =
 1 2 
 2 3 
In this example, A is a symmetric matrix.
B =
 0 1 2 
 1 0 3 
 2 3 0 
In this example, B is a skewsymmetric matrix.
Differentiate between Singular and Nonsingular Matrices
This learning outcome requires you to be able to differentiate between singular and nonsingular matrices. Matrices are important mathematical tools that are used in various fields. Understanding the properties of matrices is crucial to solving problems in mathematics, computer science, and engineering. Singular and nonsingular matrices are two important types of matrices that have distinct characteristics and properties.
To achieve this learning outcome, you should be able to:
 Define what a singular and nonsingular matrix is
 Describe the properties of singular and nonsingular matrices
 Provide examples of singular and nonsingular matrices
 Explain the significance of singular and nonsingular matrices
Let’s discuss each of these in more detail:
Definition of singular and nonsingular matrices:
 A matrix is said to be singular if its determinant is equal to zero. A matrix is said to be nonsingular if its determinant is not equal to zero. The determinant of a matrix is a scalar value that can be computed using the elements of the matrix.
 Properties of singular and nonsingular matrices:
 Singular matrices are not invertible, meaning that they cannot be transformed into their inverse matrix.
 Nonsingular matrices are invertible, meaning that they can be transformed into their inverse matrix.
 The determinant of a singular matrix is zero.
 The determinant of a nonsingular matrix is nonzero.
Examples of singular and nonsingular matrices:
Example of a singular matrix:
A =
 1 2 
 2 4 
The determinant of matrix A is equal to (14) – (22) = 0, which means that matrix A is singular.
Example of a nonsingular matrix:
B =
 1 3 
 2 5 
The determinant of matrix B is equal to (15) – (32) = 1, which means that matrix B is nonsingular.
Significance of singular and nonsingular matrices:
Singular and nonsingular matrices have important applications in mathematics, computer science, and engineering. For example, in linear algebra, nonsingular matrices play a crucial role in solving systems of linear equations. Nonsingular matrices can also be used to compute the inverse of a matrix, which is an important operation in many fields. Singular matrices, on the other hand, have limited applications, and are often avoided in computations as they do not have an inverse matrix.
Overall, understanding the properties of singular and nonsingular matrices is crucial in various mathematical and computational applications.
Describe Algebraic Operations on Matrices
This learning outcome requires you to describe algebraic operations on matrices. Matrices are used to represent data and perform calculations in many fields, such as engineering, physics, and computer science. Understanding how to perform algebraic operations on matrices is crucial for solving problems in these fields.
To achieve this learning outcome, you should be able to:
 Define what algebraic operations are on matrices
 Describe the basic operations on matrices (addition, subtraction, scalar multiplication)
 Explain the properties of algebraic operations on matrices
 Perform algebraic operations on matrices
 Provide examples of how algebraic operations on matrices can be used to solve problems
Let’s discuss each of these in more detail:
 Definition of algebraic operations on matrices:
Algebraic operations on matrices involve performing arithmetic operations on matrices. These operations include addition, subtraction, scalar multiplication, and matrix multiplication.
 Basic operations on matrices:
a. Addition: Matrices can be added if they have the same dimensions. The addition of two matrices is done elementwise, i.e., each element in the first matrix is added to the corresponding element in the second matrix. For example,
A =
 1 2 
 3 4 
B =
 5 6 
 7 8 
A + B =
 6 8 
 10 12 
b. Subtraction: Matrices can be subtracted if they have the same dimensions. The subtraction of two matrices is done elementwise, i.e., each element in the first matrix is subtracted from the corresponding element in the second matrix.
c. Scalar multiplication: A matrix can be multiplied by a scalar value. This is done by multiplying each element in the matrix by the scalar value.
Properties of algebraic operations on matrices:
a. Commutative property of addition: A + B = B + A
b. Associative property of addition: (A + B) + C = A + (B + C)
c. Distributive property of scalar multiplication: a(B + C) = aB + aC
d. Associative property of scalar multiplication: a(bA) = (ab)A
Performing algebraic operations on matrices:
Performing algebraic operations on matrices involves applying the basic operations and properties to matrices. For example, to compute the product of two matrices, we need to apply the distributive property of matrix multiplication over addition. This property states that A(B+C) = AB + AC.
Examples of how algebraic operations on matrices can be used to solve problems:
Matrix algebra is widely used in many fields, such as engineering, physics, and computer science. For example, matrix addition can be used to calculate the total distance traveled by a car, where the distances traveled in each hour are stored in a matrix. Scalar multiplication can be used to adjust the brightness of an image. Matrix multiplication can be used to represent and solve systems of linear equations.
Overall, understanding algebraic operations on matrices is essential for solving problems in many fields.
Describe Elementary operations or Transformation of a Matrix
This learning outcome requires you to describe elementary operations or transformations of a matrix. Elementary operations are simple operations that can be performed on a matrix without changing its underlying structure. These operations include row operations, column operations, and scalar operations. Understanding these operations is important for performing more complex operations on matrices and solving systems of linear equations.
To achieve this learning outcome, you should be able to:
 Define what elementary operations or transformations of a matrix are
 Describe the three types of elementary operations or transformations: row operations, column operations, and scalar operations
 Explain the effects of each of these operations on a matrix
 Perform elementary operations or transformations on a matrix
 Provide examples of how elementary operations or transformations can be used to solve problems
Let’s discuss each of these in more detail:
 Definition of elementary operations or transformations of a matrix:
Elementary operations or transformations of a matrix involve simple operations that can be performed on the matrix without changing its underlying structure. These operations include row operations, column operations, and scalar operations.
Three types of elementary operations or transformations:
a. Row operations: These are operations that are performed on the rows of a matrix. There are three types of row operations:

 Interchange two rows
 Multiply a row by a nonzero scalar
 Add a multiple of one row to another row
b. Column operations: These are operations that are performed on the columns of a matrix. There are three types of column operations:
– Interchange two columns
– Multiply a column by a nonzero scalar
– Add a multiple of one column to another column
c. Scalar operations: These are operations that are performed on a single element of a matrix. There are two types of scalar operations:
– Multiply an element by a nonzero scalar
– Add a multiple of one element to another element
Effects of each of these operations on a matrix:
a. Row operations: Row operations change the row space of the matrix. The row space is the space spanned by the rows of the matrix.
b. Column operations: Column operations change the column space of the matrix. The column space is the space spanned by the columns of the matrix.
c. Scalar operations: Scalar operations change the entries of the matrix.
Performing elementary operations or transformations on a matrix:
Performing elementary operations or transformations on a matrix involves applying the desired operation to the matrix. For example, to perform a row operation on a matrix, we need to perform the desired operation on one or more of the rows of the matrix.
Examples of how elementary operations or transformations can be used to solve problems:
Elementary operations or transformations can be used to solve many problems, such as finding the inverse of a matrix, solving systems of linear equations, and finding the rank of a matrix. For example, row operations can be used to transform a matrix into row echelon form, which can then be used to solve systems of linear equations. Column operations can be used to simplify a matrix and make it easier to work with. Scalar operations can be used to adjust the values of a matrix to better fit a particular problem.
Overall, understanding elementary operations or transformations of a matrix is important for performing more complex operations on matrices and solving problems in many fields.
Calculate the Inverse using Elementary operations
This learning outcome requires you to calculate the inverse of a matrix using elementary operations. The inverse of a matrix is an important concept in linear algebra and is used in many applications, including solving systems of linear equations, finding determinants, and more. The inverse of a matrix is a matrix that, when multiplied by the original matrix, produces the identity matrix.
To achieve this learning outcome, you should be able to:
 Define what the inverse of a matrix is
 Understand when a matrix is invertible
 Describe how to calculate the inverse of a matrix using elementary operations
 Perform the necessary elementary operations to calculate the inverse of a matrix
 Verify that the calculated inverse is correct by multiplying it by the original matrix
Let’s discuss each of these in more detail:
Definition of the inverse of a matrix
The inverse of a matrix is a matrix that, when multiplied by the original matrix, produces the identity matrix. In other words, if A is an n x n matrix, and if there exists a matrix B such that AB = BA = I (the identity matrix), then B is the inverse of A.
When a matrix is invertible?
A matrix is invertible (or nonsingular) if and only if its determinant is nonzero. If the determinant of a matrix is zero, then the matrix is not invertible (or singular).
How to calculate the inverse of a matrix using elementary operations?
To calculate the inverse of a matrix using elementary operations, we need to perform row operations on the original matrix until it is transformed into the identity matrix. We then perform the same row operations on the identity matrix to get the inverse of the original matrix. The row operations we use to transform the original matrix into the identity matrix must be carefully chosen to avoid changing the value of the determinant.
Performing the necessary elementary operations to calculate the inverse of a matrix:
To calculate the inverse of a matrix, we perform the following steps:
a. Augment the original matrix with the identity matrix to form a new matrix.
b. Perform row operations on the new matrix to transform the original matrix into the identity matrix. The same row operations must be performed on the identity matrix as well.
c. The resulting matrix is the inverse of the original matrix.
Verifying that the calculated inverse is correct by multiplying it by the original matrix:
To verify that the calculated inverse is correct, we can multiply it by the original matrix. If the result is the identity matrix, then the calculated inverse is correct.
Example:
Let’s say we have the following 3×3 matrix A:
A=
 1 2 3 
 4 5 6 
 7 8 9 
To calculate the inverse of A, we first need to check if it is invertible by calculating its determinant:
det(A) = 1*(5*98*6) – 2*(4*97*6) + 3*(4*85*7) = 0
Since the determinant of A is zero, it is not invertible.
Now, let’s consider a different 3×3 matrix B:
 1 0 1 
 1 1 0 
 0 1 1 
To calculate the inverse of B, we need to perform the following steps:
To find the inverse of a matrix B using elementary row operations, we can augment B with the identity matrix and perform row operations until the lefthand side becomes the identity matrix. The righthand side will then be the inverse of B.
Let’s start by writing B and the identity matrix side by side:
 1 0 1  1 0 0 
 1 1 0  0 1 0 
 0 1 1  0 0 1 
We can perform row operations on the augmented matrix to transform the lefthand side into the identity matrix:
 Subtract the first row from the second row:
 1 0 1  1 0 0 
 0 1 1 1 1 0 
 0 1 1  0 0 1 
 Subtract the second row from the third row:
 1 0 1  1 0 0 
 0 1 1 1 1 0 
 0 0 2  1 1 1 
 Divide the third row by 2:
 1 0 1  1 0 0 
 0 1 1 1 1 0 
 0 0 1  1/2 1/2 1/2 
 Add the third row to the second row:
 1 0 1  1 0 0 
 0 1 0  1/2 1/2 1/2 
 0 0 1  1/2 1/2 1/2 
 Subtract the third row from the first row:
 1 0 0  1/2 1/2 1/2 
 0 1 0  1/2 1/2 1/2 
 0 0 1  1/2 1/2 1/2 
The lefthand side is now the identity matrix, so the righthand side is the inverse of B:
B^{1} =
 1/2 1/2 1/2 
 1/2 1/2 1/2 
 1/2 1/2 1/2 
Describe and Calculate Transpose of a Matrix
The transpose of a matrix is a new matrix formed by interchanging the rows and columns of the original matrix. The transpose of a matrix can be useful in many areas of mathematics and science, including linear algebra, statistics, and physics. In this learning outcome, we will explore the definition of matrix transpose and how to calculate it.
Definition:
The transpose of a matrix A, denoted by A^{T}, is obtained by interchanging the rows and columns of matrix A. More formally, if A = [aij] is an m x n matrix, then the transpose of A is an n x m matrix, denoted by A^{T}, with entries given by A^{T} = [bij], where bij = aji.
Example:
Consider the following matrix:
A = [1 2 3
4 5 6]
To find the transpose of A, we simply interchange the rows and columns of A to get:
A^{T} = [1 4
2 5
3 6]
Notice that the original matrix A was a 2 x 3 matrix, but its transpose A^{T} is a 3 x 2 matrix. Also, the elements in the first row of A became the elements in the first column of A^{T,} and similarly, the elements in the second row of A became the elements in the second column of A^{T}.
Calculation:
To calculate the transpose of a matrix, we simply need to interchange the rows and columns of the original matrix. For example, let us consider the matrix A = [a_{ij}], which is an m x n matrix. To find the transpose of A, we follow these steps:
 Create a new matrix A^{T} with dimensions n x m, where n is the number of rows in A, and m is the number of columns in A.
 For each element aij in the original matrix A, copy it to the corresponding element in the new matrix A^{T}, but with the indices i and j swapped. That is, bij = aji for all i and j.
Example:
Let us consider the matrix
A = [1 2 3
4 5 6
7 8 9]
To find the transpose of A, we follow the steps mentioned above:
 Create a new matrix A^{T} with dimensions 3 x 3.
 For each element aij in the original matrix A, copy it to the corresponding element in the new matrix A^{T}, but with the indices i and j swapped. That is, b11 = a11, b12 = a21, b13 = a31, b21 = a12, b22 = a22, b23 = a32, b31 = a13, b32 = a23, b33 = a33.
After performing the above steps, we get the transpose of A as follows:
A^{T} = [1 4 7
2 5 8
3 6 9]
Conclusion:
In this learning outcome, we explored the definition of matrix transpose and how to calculate it. We saw that the transpose of a matrix is obtained by interchanging the rows and columns of the original matrix. We also saw how to calculate the transpose of a matrix by simply interchanging the rows and columns of the original matrix. The transpose of a matrix is an important concept in linear algebra and is used in many areas of mathematics and science.
Describe Symmetric and SkewSymmetric Matrices
Matrices are widely used in various fields such as mathematics, engineering, physics, and computer science. In linear algebra, matrices can be classified into different types based on their properties. Two such types of matrices are symmetric and skewsymmetric matrices. In this learning outcome, we will explore the definitions of these matrices and their properties.
Symmetric Matrix:
A symmetric matrix is a square matrix that is equal to its transpose. Formally, an n x n matrix A is symmetric if and only if A^{T} = A. In other words, the elements above and below the main diagonal are reflections of each other. For example, consider the following matrix:
A = [2 3 4
3 5 6
4 6 7]
This matrix is symmetric because the elements above and below the main diagonal are reflections of each other. The transpose of A is:
A^{T} = [2 3 4
3 5 6
4 6 7]
Since A^{T} = A, the matrix A is symmetric.
Properties of Symmetric Matrices:
 The diagonal entries of a symmetric matrix are always real.
 The eigenvalues of a symmetric matrix are always real.
 The eigenvectors of a symmetric matrix are always orthogonal.
SkewSymmetric Matrix:
A skewsymmetric matrix is a square matrix that is equal to the negative of its transpose. Formally, an n x n matrix A is skewsymmetric if and only if A^{T} = A. In other words, the elements above and below the main diagonal are negatives of each other. For example, consider the following matrix:
A = [0 2 3
2 0 4
3 4 0]
This matrix is skewsymmetric because the elements above and below the main diagonal are negatives of each other. The transpose of A is:
A^{T} = [0 2 3
2 0 4
3 4 0]
Since A^{T} = A, the matrix A is skewsymmetric.
Properties of SkewSymmetric Matrices:
 The diagonal entries of a skewsymmetric matrix are always zero.
 The eigenvalues of a skewsymmetric matrix are always pure imaginary or zero.
 The eigenvectors of a skewsymmetric matrix corresponding to distinct eigenvalues are always orthogonal.
Example:
Let us consider the following matrix:
A = [1 2 3
4 5 6
7 8 9]
To determine whether A is symmetric or skewsymmetric, we need to calculate its transpose. The transpose of A is:
A^{T} = [1 4 7
2 5 8
3 6 9]
Since A is not equal to its transpose A^{T} and A is not equal to the negative of its transpose, we can conclude that A is neither symmetric nor skewsymmetric.
Conclusion:
In this learning outcome, we explored the definitions of symmetric and skewsymmetric matrices and their properties. We saw that a symmetric matrix is a square matrix that is equal to its transpose, while a skewsymmetric matrix is a square matrix that is equal to the negative of its transpose. We also saw some of the important properties of these matrices, including the reality of eigenvalues and orthogonality of eigenvectors for symmetric matrices, and the zero diagonal entries and pure imaginary or zero eigenvalues for skewsymmetric matrices. Understanding these properties can be useful in solving various problems.
Describe Orthogonal Matrix and its properties
An orthogonal matrix is a square matrix whose columns and rows are orthonormal. In other words, the dot product of any two distinct columns or rows is zero, and the dot product of any column or row with itself is one. In this learning outcome, we will explore the definition of an orthogonal matrix and its properties.
Orthogonal Matrix:
A square matrix A is orthogonal if and only if its transpose A^{T} is equal to its inverse A^{(1)}. Formally, A is orthogonal if and only if A^{T} A = AA^{T} = I, where I is the identity matrix. This implies that the columns and rows of A are orthonormal. For example, consider the following matrix:
A = [1/sqrt(2) 1/sqrt(2)
1/sqrt(2) 1/sqrt(2)]
The columns of A are orthonormal because their dot product is zero, and their magnitude is one. Similarly, the rows of A are orthonormal because their dot product is zero, and their magnitude is one. The transpose of A is:
A^{T} = [1/sqrt(2) 1/sqrt(2)
1/sqrt(2) 1/sqrt(2)]
We can verify that A is orthogonal by checking if A^{T} A = AA^{T} = I:
A^{T} A = [1 0
0 1]
AA^{T} = [1 0
0 1]
Since A^{T} A = AA^{T} = I, we can conclude that A is orthogonal.
Properties of Orthogonal Matrices:
 The determinant of an orthogonal matrix is either 1 or 1.
 The inverse of an orthogonal matrix is its transpose.
 The columns and rows of an orthogonal matrix are orthonormal.
 The product of two orthogonal matrices is also an orthogonal matrix.
 The transpose of a product of orthogonal matrices is the product of their transposes in reverse order.
Example:
Let us consider the following matrix:
A = [0 1
1 0]
To determine whether A is orthogonal, we need to calculate its transpose and its inverse. The transpose of A is:
AA^{T} = [0 1
1 0]
To calculate the inverse of A, we need to solve the equation AX = I, where I is the identity matrix. We have:
[0 1  1 0]
[1 0  0 1]
Performing row operations, we obtain:
[1 0  0 1]
[0 1  1 0]
Therefore, the inverse of A is:
A^{(1)} = [0 1
1 0]
We can now verify if A is orthogonal by checking if A^{T} A = AA^{T} = I:
A^{T}A = [1 0
0 1]
AA^{T} = [1 0
0 1]
Since A^{T} A = AA^{T} = I, we can conclude that A is orthogonal.
Conclusion:
In this learning outcome, we explored the definition of an orthogonal matrix and its properties. We saw that an orthogonal matrix is a square matrix whose columns and rows are orthonormal. We also saw some of the important properties of orthogonal matrices, including the orthonormality of its columns and rows, the inverse of an orthogonal matrix being its transpose, and the product of two orthogonal matrices being an orthogonal matrix.
Describe Complex conjugate of a Matrix
A complex conjugate of a number is the number with the same real part and an opposite imaginary part. Similarly, a complex conjugate of a matrix is the matrix with the same real part and an opposite imaginary part for each element. In this learning outcome, we will explore the definition of the complex conjugate of a matrix and how to calculate it.
Complex Conjugate of a Matrix:
Let A = [a_{ij}] be a matrix of complex numbers. The complex conjugate of A, denoted by A*, is the matrix obtained by taking the complex conjugate of each element of A. Formally, A* = [a_{ij*}], where a_{ij*} is the complex conjugate of a_{ij}.
For example, let A be the following matrix of complex numbers:
A = [2+3i 45i
1+i 0]
To calculate the complex conjugate of A, we need to take the complex conjugate of each element of A. Thus, we obtain:
A* = [23i 4+5i
1i 0]
Note that each element of A* is the complex conjugate of the corresponding element in A.
Properties of Complex Conjugate of a Matrix:
 The complex conjugate of a matrix is distributive with respect to addition: (A + B)* = A* + B*.
 The complex conjugate of a matrix is distributive with respect to scalar multiplication: (kA)* = kA*, where k is a scalar.
 The complex conjugate of a product of matrices is the product of the complex conjugates in reverse order: (AB)* = B* A*.
Example:
Let us consider the following matrices:
A = [1+i 23i
3 4+2i]
B = [5i 1+2i
0 1i]
To calculate the complex conjugate of A, we need to take the complex conjugate of each element of A. Thus, we obtain:
A* = [1i 2+3i
3 42i]
Similarly, to calculate the complex conjugate of B, we need to take the complex conjugate of each element of B. Thus, we obtain:
B* = [5+i 12i
0 1+i]
We can now verify the properties of the complex conjugate of a matrix:
 The complex conjugate of the sum of matrices A and B is equal to the sum of their complex conjugates:
(A + B)* = [6 3 i
3+i 3+i]
A* + B* = [6 3 i
3+i 3+i]
 The complex conjugate of the product of the scalar k and matrix A is equal to the product of the scalar k and the complex conjugate of A:
(kA)* = [k(1i) k(2+3i)
k(3) k(42i)]
kA* = [k(1+i) k(23i)
k(3) k(4+2i)]
3. The complex conjugate of the product of two matrices is not equal to the product of their complex conjugates in reverse order.
Let A and B be complex matrices of compatible sizes, then the complex conjugate of the product AB is given by taking the complex conjugate of each element and reversing the order of the matrices, i.e.,
(AB)* = B* A*
where * denotes complex conjugation.
On the other hand, the product of the complex conjugate of B and the complex conjugate of A is given by
B* A*
Therefore, in general, (AB)* ≠ B* A*. However, if either A or B is a real matrix (i.e., all of its elements are real), then the two expressions are equal, since taking the complex conjugate of a real number does not change it.
Describe Hermitian and Skew Hermitian of a Matrix and its properties
Hermitian Matrix:
A Hermitian matrix is a square matrix that is equal to its conjugate transpose. In other words, a matrix A is Hermitian if and only if A = A* , where A* is the conjugate transpose of A.
Mathematically, for a square matrix A with entries a_{ij}, the conjugate transpose of A denoted by A* is obtained by taking the transpose of A and then taking the complex conjugate of each entry. That is, (A*)_{ij} = (conj(A_{ij}))^{T}, where conj() represents the complex conjugate.
Properties of Hermitian Matrices:
 The diagonal entries of a Hermitian matrix are real numbers.
 The eigenvalues of a Hermitian matrix are real numbers.
 The eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal.
Example:
Consider the matrix A = [2 + i, 3 – 2i; 3 + 2i, 4 – i].
To check whether A is Hermitian, we need to check whether A = A*.
Taking the conjugate transpose of A, we get A* = [2 – i, 3 + 2i; 3 – 2i, 4 + i].
Comparing A and A*, we can see that A is indeed Hermitian.
Skew Hermitian Matrix:
A skewHermitian matrix is a square matrix that is equal to the negation of its conjugate transpose. In other words, a matrix A is skewHermitian if and only if A = A* , where A* is the conjugate transpose of A.
Mathematically, for a square matrix A with entries a_{ij,} the conjugate transpose of A denoted by A* is obtained by taking the transpose of A and then taking the complex conjugate of each entry. That is, (A*)_{ij} = (conj(A_{ij}))^{T}, where conj() represents the complex conjugate.
Properties of Skew Hermitian Matrices:
 The diagonal entries of a skew Hermitian matrix are purely imaginary numbers.
 The eigenvalues of a skew Hermitian matrix are purely imaginary or zero.
 The eigenvectors corresponding to distinct eigenvalues of a skew Hermitian matrix are orthogonal.
Example:
Consider the matrix A = [0, 2 + i; 2 + i, 0].
To check whether A is skewHermitian, we need to check whether A = A*.
Taking the conjugate transpose of A, we get A* = [0, 2 – i; 2 – i, 0].
Negating A*, we get A* = [0, 2 + i; 2 + i, 0].
Comparing A and A*, we can see that A is indeed skewHermitian.
Describe Unitary Matrix with example
A unitary matrix is a complex square matrix whose conjugate transpose is its inverse. In other words, a matrix U is unitary if and only if UU = UU = I, where I is the identity matrix and U* is the conjugate transpose of U.
Mathematically, for a square matrix U with entries uij, the conjugate transpose of U denoted by U* is obtained by taking the transpose of U and then taking the complex conjugate of each entry. That is, (U*)ij = (conj(Uij))^{T}, where conj() represents the complex conjugate.
Properties of Unitary Matrices:
 Unitary matrices are normpreserving. That is, Ux = x for any vector x, where . denotes the Euclidean norm.
 The eigenvalues of a unitary matrix U have absolute value 1, and the eigenvectors corresponding to distinct eigenvalues of U are orthogonal.
 The determinant of a unitary matrix U has absolute value 1, that is, det(U) = 1.
Example:
Consider the matrix U = [1/sqrt(2), 1/sqrt(2); 1/sqrt(2), 1/sqrt(2)].
To check whether U is unitary, we need to check whether UU = UU = I.
Taking the conjugate transpose of U, we get U* = [1/sqrt(2), 1/sqrt(2); 1/sqrt(2), 1/sqrt(2)].
Multiplying U and U* and comparing with the identity matrix I, we get:
UU* = [1, 0; 0, 1]
And
U*U = [1, 0; 0, 1]
Hence, U is indeed unitary.
We can also check that U is normpreserving by checking that Ux = x for any vector x.
Consider the vector x = [a, b]. Then, we have
Ux^{2} = [a/sqrt(2) + b/sqrt(2), a/sqrt(2) + b/sqrt(2)]^{2}
= (a/sqrt(2) + b/sqrt(2))^{2} + (a/sqrt(2) + b/sqrt(2))^{2}
= a^{2 }+ b^{2}
= x^{2}
Hence, U is normpreserving.
Describe Determinant and its properties
The determinant is a scalar value that can be calculated for a square matrix. It is used to determine whether a matrix is invertible, and it also has several other important applications in linear algebra, such as computing eigenvalues and eigenvectors.
The determinant of an n x n matrix A is denoted by det(A) or A, and it is defined recursively as follows:
For a 1 x 1 matrix A, det(A) = a11.
For an n x n matrix A, where n > 1, det(A) is defined as the sum of the products of the elements in any row or column of A, each multiplied by its corresponding cofactor. The cofactor of an element aij in A is defined as (1)^{(i+j)} times the determinant of the (n1) x (n1) matrix obtained by deleting the ith row and jth column of A. That is, Cij = (1)^{(i+j)} det(Aij), where Aij is the (n1) x (n1) matrix obtained by deleting the ith row and jth column of A.
Properties of the determinant:
 If A is a triangular matrix, then det(A) is the product of the entries on the main diagonal of A.
 If A is invertible, then det(A) is nonzero. If A is not invertible, then det(A) is zero.
 If A and B are two matrices of the same size, then det(AB) = det(A) det(B).
 If A is a matrix and k is a scalar, then det(kA) = k^{n} det(A), where n is the size of A.
 If A is a matrix, then det(A^{T}) = det(A), where A^{T} is the transpose of A.
 If A and B are similar matrices, then they have the same determinant. That is, det(A) = det(B) for any matrices A and B that are similar.
Example:
Consider the matrix A = [1 2; 3 4]. To find its determinant, we can use the recursive definition and choose the first row to calculate the determinant. Thus, we have
det(A) = 1 * (1)^{(1+2) }* det([3 4]) – 2 * (1)^{(1+3) }* det([2 4]) + 1 * (1)^{(1+4) }* det([2 3])
= 1 * (1) * (3 * 4 – 4 * 0) – 2 * (1) * (2 * 4 – 4 * 3) + 1 * (1 * 4 – 3 * 2)
= 2
Therefore, the determinant of A is 2. Since det(A) is nonzero, we can conclude that A is invertible. We can also check that the determinant is preserved under transpose, i.e., det(A^{T}) = det(A). This is because A and A^{T} are identical matrices in this case.
We can also verify that the property det(AB) = det(A) det(B) holds for A and its inverse matrix. Since A is invertible, we have det(A) != 0, and the inverse matrix of A is given by A^{(1)} = 1/(2) * [4 2; 3 1]. Therefore, we have det(AA^{(1)}) = det(I) = 1, and det(A)
Describe and apply Laplace’s Method of Expansion of Determinant
Laplace’s method of expansion, also known as cofactor expansion or minor expansion, is a technique for computing the determinant of a matrix by recursively expanding along a row or column. The method is named after the French mathematician PierreSimon Laplace, who first described it in the late 18th century.
Suppose we have an n x n matrix A. We can expand the determinant of A along the ith row or jth column using the formula:
det(A) = a(i,1) C(i,1) + a(i,2) C(i,2) + … + a(i,n) C(i,n) = sum(a(i,j) C(i,j), j=1 to n)
or
det(A) = a(1,j) C(1,j) + a(2,j) C(2,j) + … + a(n,j) C(n,j) = sum(a(i,j) C(i,j), i=1 to n)
where a(i,j) is the (i,j)th element of A, and C(i,j) is the corresponding cofactor of a(i,j). The cofactor C(i,j) is defined as (1)^{(i+j)} times the determinant of the (n1) x (n1) matrix obtained by deleting the ith row and jth column of A.
The Laplace expansion formula can be applied recursively by choosing any row or column and expanding along it. This reduces the size of the matrix by one and allows us to compute the cofactors of the remaining elements. The process is repeated until we obtain a 1 x 1 matrix, for which the determinant is simply the single element.
The advantage of Laplace’s method is that it does not require us to compute the inverse of the matrix, and it can be used to compute the determinant of any matrix, regardless of whether it is invertible or not. However, it can be computationally expensive for large matrices, as the number of cofactors that need to be computed grows exponentially with the size of the matrix.
Example:
Let us consider the matrix A = [1 2 3; 4 5 6; 7 8 9]. To find its determinant using Laplace’s method, we can expand along the first row as follows:
det(A) = 1 * C(1,1) – 2 * C(1,2) + 3 * C(1,3)
where C(1,1), C(1,2), and C(1,3) are the cofactors of the elements 1, 2, and 3, respectively. To compute the cofactors, we need to find the determinants of the 2 x 2 matrices obtained by deleting the first row and the corresponding column. We have:
C(1,1) = (1)^{(1+1)} det([5 6; 8 9]) = (1) det([5 6; 8 9]) = 3
C(1,2) = (1)^{(1+2)} det([4 6; 7 9]) = det([4 6; 7 9]) = 6
C(1,3) = (1)^{(1+3)} det([4 5; 7 8]) = (1) det([4 5; 7 8]) = 3
Therefore, we have:
det(A) = 1 * (3) – 2 * (6) + 3 *
Describe Adjoint of a Matrix and its Properties
The adjoint of a square matrix A, denoted by adj(A), is a matrix whose elements are the determinants of the minors of A with signs according to a checkerboard pattern. More precisely, if A = [a_{{ij}}] is an nxn matrix, then the (i, j)^{th }entry of adj(A) is given by:
adj(A)_{{ji} }= (1){i+j} det(A{ji})
where A_{{ji}} is the (n1)×(n1) submatrix of A obtained by deleting the ith row and the jth column of A.
Some properties of the adjoint matrix include:
 If A is invertible, then adj(A) = (1/A)A^{{1}}, where A is the determinant of A.
 If A is not invertible, then adj(A) is not invertible either.
 adj(adj(A)) = A^{(n2)}A, where n is the dimension of A.
 If A and B are nxn matrices, then adj(AB) = adj(B)adj(A).
 If A is a symmetric matrix, then adj(A) = det(A)A^{{1}.}
 If A is a skewsymmetric matrix, then adj(A) = (1)^{ (n1)}A^{{n1}}, where n is the dimension of A.
 If A is a Hermitian matrix (i.e., a complex symmetric matrix), then adj(A) = det(A)A^{{1}}.
 If A is a unitary matrix (i.e., AA^{H} = A^{HA }= I, where A^{H} is the conjugate transpose of A), then adj(A) = A^{H}.
These properties can be useful in various applications, such as solving systems of linear equations, computing determinants, and finding inverses of matrices.
Describe Adjoint of a Determinant
The adjoint of a determinant is a matrix which is obtained by taking the transpose of the matrix of cofactors of the given determinant. The adjoint of a determinant has several properties which are useful in many applications.
Properties of Adjoint of a Determinant:
 Multiplication: Let A be an n x n matrix and let B be the matrix obtained by replacing the ith row and jth column of A by a vector u. Then det(B) = det(A) + u^{T} adj(A) e_{j}, where e_{j }is the j^{th} unit vector.
Example:
Let A =
2 3 5
1 4 6
7 8 9
The determinant of A is A = 18
Let B be the matrix obtained by replacing the 2nd row and 3rd column of A by the vector (1, 2, 3). Then
B =
2 3 5
1 4 1
7 8 3
The matrix of cofactors of A is
8 6 2
2 3 2
2 2 2
The transpose of the matrix of cofactors of A is
8 2 2
6 3 2
2 2 2
The vector u = (1, 2, 3) and the 3rd unit vector e_{3} = (0, 0, 1). Therefore, we can use the formula
det(B) = det(A) + u^{T} adj(A) e_{j}
to compute the determinant of B as follows:
u^{T} adj(A) e_{j} = (1, 2, 3) *
(8, 2, 2)
(6, 3,2)
(2, 2, 2)
 (0, 0, 1)^{T}
= 10
Therefore, det(B) = det(A) + u^{T} adj(A) e_{j }= 18 – 10 = 28.
 Inverse: Let A be an invertible n x n matrix. Then the inverse of A can be expressed as A^{{1}} = (1/A) adj(A), where A is the determinant of A.
Example:
Let A =
1 2
3 4
The determinant of A is A = 2, which is nonzero, therefore A is invertible.
The matrix of cofactors of A is
4 2
3 1
The transpose of the matrix of cofactors of A is
4 3
2 1
Therefore, adj A =
4 3
2 1
The inverse of A is given by
A^{{1}} = (1/A) adj(A) = (1/2) *
(4 3)
(2 1)
= 2/2 3/2
1/2 1/2
Therefore, A^{{1}} =
1 1
1.5 0.5
Apply Adjoint to find Inverse of a Matrix
One way to find the inverse of a matrix A is to use its adjoint matrix adj(A) and its determinant A. The inverse of A is given by:
A^{{1} }= (1/A) adj(A)
where adj(A) is the adjoint of A, and A is the determinant of A.
To compute the adjoint matrix adj(A), we need to compute the determinant of each of the minors of A and put them in the appropriate positions of the adjoint matrix, with signs determined by a checkerboard pattern.
Here is an example of how to find the inverse of a 3×3 matrix A using the adjoint method:
Suppose we have the matrix:
A = [2 1 3
0 1 2
1 0 1]
First, we need to compute the determinant of A:
A = 2(11 – 20) – 1(01 – 21) + 3(00 – 11) = 1
Next, we need to compute the matrix of minors M of A, which is obtained by computing the determinants of the 2×2 matrices obtained by deleting each row and column of A:
M = [1 2 1
3 2 1
1 2 1]
Then, we need to compute the matrix of cofactors C, which is obtained by multiplying each element of M by the appropriate sign according to a checkerboard pattern:
C = [1 2 1
3 2 1
1 2 1]
Finally, we can compute the adjoint matrix adj(A) by taking the transpose of C:
adj(A) = [1 3 1
2 2 2
1 1 1]
Now we can use the formula for the inverse of A:
A^{{1}} = (1/A) adj(A) = (1/1) [1 3 1; 2 2 2; 1 1 1] = [1 3 1; 2 2 2; 1 1 1]
Therefore, the inverse of A is:
A^{{1} }= [1 3 1
2 2 2
1 1 1]
Note that we can check that A A^{{1}} = A^{{1}} A = I, where I is the identity matrix, to confirm that our calculation is correct.
Describe Linear Simultaneous Equations
Linear simultaneous equations are a set of equations in which each equation is a linear equation in the same set of variables. These equations are called simultaneous because they are considered together and are usually solved for a unique solution of the variables that satisfies all the equations in the set.
The general form of a system of linear simultaneous equations is:
a_{{11}}x_{1} + a_{{12}}x_{2 }+ … + a_{{1n}}x_{n} = b_{1}
a_{{21}}x_{1} + a_{{22}}x_{2} + … + a_{{2n}}x_{n }= b_{2}
…
a_{{m1}}x_{1 }+ a_{{m2}}x_{2} + … + a_{{mn}}x_{n }= b
_{m}
where x_{1}, x_{2}, …, x_{n} are the variables we want to solve for, a_{{ij}} are the coefficients of the variables, b_{i} are the constant terms, and m is the number of equations in the system.
The solution to a system of linear simultaneous equations is a set of values for the variables x_{1}, x_{2}, …, x_{n} that satisfy all the equations in the system. There are three possible outcomes for the solution of a system of linear simultaneous equations:
 The system has a unique solution, which means that there is only one set of values for the variables that satisfies all the equations in the system.
 The system has infinitely many solutions, which means that there are infinitely many sets of values for the variables that satisfy all the equations in the system.
 The system has no solution, which means that there is no set of values for the variables that satisfies all the equations in the system.
To solve a system of linear simultaneous equations, we can use various methods such as elimination, substitution, and matrix methods. These methods involve manipulating the equations to obtain a simpler system of equations that can be easily solved for the variables.
Describe the steps to calculate the solutions of Linear Simultaneous Equations using Matrix Inversion Method
Matrix inversion is a powerful technique used to solve linear simultaneous equations. It is a process of finding the inverse of a matrix, which is then used to solve systems of linear equations. The matrix inversion method is based on the concept of the inverse of a matrix, which is a matrix that, when multiplied by the original matrix, gives the identity matrix as the result. In this method, the equations are represented in the form of a matrix, and the inverse of the matrix is calculated to obtain the solution of the system of equations.
Steps to calculate the solutions of Linear Simultaneous Equations using Matrix Inversion method:
Step 1: Formulate the system of equations in matrix form:
The first step in using the matrix inversion method to solve linear simultaneous equations is to represent the equations in the form of a matrix. Let’s consider a system of two linear equations with two variables, as shown below:
2x + y = 5
x – y = 1
The above system of equations can be represented in matrix form as:
[2 1] [x] = [5]
[1 1] [y] [1]
Step 2: Find the inverse of the coefficient matrix:
The next step is to find the inverse of the coefficient matrix. The inverse of a matrix is denoted by A^{1}, and it is the matrix that, when multipliey the original matrix A, gives the identity matrix I as the result. The inverse of a matrix can be found using various methods, such as GaussJordan elimination or the adjugate method. Let’s assume that the inverse of the coefficient matrix [2 1; 1 1] is [1/3 1/3; 1/3 2/3].
Step 3: Multiply the inverse of the coefficient matrix with the constant matrix:
The third step is to multiply the inverse of the coefficient matrix with the constant matrix [5; 1]. The product of these matrices gives the solution to the system of equations. The product can be calculated as:
[1/3 1/3; 1/3 2/3] [5] = [2]
[1] [1]
Therefore, the solution to the system of equations is x=2, y=1.
Example:
Let’s consider another system of equations as follows:
3x + 2y + z = 9
2x – y + 4z = 7
x + y – 2z = 4
This system of equations can be represented in matrix form as:
[3 2 1] [x] = [9]
[2 1 4] [y] [7]
[1 1 2] [z] [4]
To solve this system of equations using the matrix inversion method, we need to follow the three steps mentioned above. First, we represent the system of equations in matrix form. Second, we find the inverse of the coefficient matrix [3 2 1; 2 1 4; 1 1 2]. Let’s assume that the inverse of the coefficient matrix is [2 1 1; 2 2 1; 3 2 0]. Finally, we multiply the inverse of the coefficient matrix with the constant matrix [9; 7; 4]. The product of these matrices gives the solution to the system of equations, which is x=1, y=2, z=3.
Describe the steps to calculate the solutions of Linear Simultaneous Equations using Cramer’s rule
Cramer’s rule is a method used to solve linear simultaneous equations. It is named after Gabriel Cramer, a Swiss mathematician who developed the rule in the mid18th century. Cramer’s rule uses the determinants of matrices to find the solution to a system of linear equations. In this method, the equations are represented in the form of matrices, and the determinants of these matrices are used to find the solution of the system of equations.
Steps to calculate the solutions of Linear Simultaneous Equations using Cramer’s rule:
Step 1: Formulate the system of equations in matrix form:
The first step in using Cramer’s rule to solve linear simultaneous equations is to represent the equations in the form of a matrix. Let’s consider a system of two linear equations with two variables, as shown below:
2x + y = 5
x – y = 1
The above system of equations can be represented in matrix form as:
[2 1] [x] = [5]
[1 1] [y] [1]
Step 2: Find the determinant of the coefficient matrix:
The second step is to find the determinant of the coefficient matrix [2 1; 1 1]. The determinant of a matrix is denoted by A, and it is a scalar v
lue that can be calculated using various methods, such as cofactor expansion or row reduction. The determinant of the coefficient matrix can be calculated as:
A = 2(1) – 1(1) = 3
Step 3: Find the determinants of the matrices obtained by replacing the columns of the coefficient matrix with the constant matrix:
The third step is to find the determinants of the matrices obtained by replacing the columns of the coefficient matrix with the constant matrix [5; 1]. To do this, we replace the first column of the coefficient matrix with the constant matrix to get the matrix [5 1; 1 1]. The determinant of this matrix can be calculated as:
x = 5(1) – 1(1) = 6
Similarly, we replace the second column of the coefficient matrix with the constant matrix to get the matrix [2 5; 1 1]. The determinant of this matrix can be calculated as:
y = 2(1) – 5(1) = 3
Step 4: Calculate the solutions of the system of equations:
The fourth and final step is to calculate the solutions of the system of equations using the determinants obtained in Step 3. The solutions can be calculated as:
x = x / A = 6 / 3 = 2
y = y / A = 3 / 3 = 1
Therefore, the solution to the system of equations is x=2, y=1.
Example:
Let’s consider another system of equations as follows:
3x + 2y + z = 9
2x – y + 4z = 7
x + y – 2z = 4
This system of equations can be represented in matrix form as:
[3 2 1] [x] = [9]
[2 1 4] [y] [7]
[1 1 2] [z] [4]
To solve this system of equations using Cramer’s rule, we need to follow the four steps mentioned above.
Describe the systems of Homogeneous and Nonhomogeneous equations
A system of linear equations is a set of two or more linear equations with the same variables. A linear equation is an equation in which the highest power of the variable is one. A system of linear equations can be classified into two categories: homogeneous and nonhomogeneous.
Homogeneous System of Equations
A system of linear equations is said to be homogeneous if all the constants on the righthand side of the equations are equal to zero. In other words, a system of equations is homogeneous if it can be represented in the form Ax = 0, where A is the coefficient matrix, x is the column vector of variables, and 0 is the column vector of zeros.
Example:
Consider the following system of equations:
x – y + 2z = 0
2x + y – 3z = 0
3x + 2y – 4z = 0
This system of equations can be written in the form Ax = 0, where:
A = [1 1 2; 2 1 3; 3 2 4]
x = [x; y; z]
0 = [0; 0; 0]
This system of equations is homogeneous because all the constants on the righthand side of the equations are zero.
Nonhomogeneous System of Equations
A system of linear equations is said to be nonhomogeneous if it has constants on the righthand side of the equations that are not equal to zero. In other words, a system of equations is nonhomogeneous if it can be represented in the form Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants.
Example:
Consider the following system of equations:
x + 2y + z = 4
2x – y + 3z = 8
3x + y – 2z = 2
This system of equations can be written in the form Ax = b, where:
A = [1 2 1; 2 1 3; 3 1 2]
x = [x; y; z]
b = [4; 8; 2]
This system of equations is nonhomogeneous because it has constants on the righthand side of the equations that are not equal to zero.
Key Differences:
The key difference between homogeneous and nonhomogeneous systems of equations is that a homogeneous system always has a trivial solution, while a nonhomogeneous system may or may not have a solution. In a homogeneous system, the solution is always x = 0, where x is the column vector of variables. In a nonhomogeneous system, the solution may be unique, have infinitely many solutions or be inconsistent depending on the values of the constants on the righthand side of the equations. Additionally, the determinant of the coefficient matrix of a homogeneous system is always zero, while the determinant of the coefficient matrix of a nonhomogeneous system may or may not be zero.
Describe the Consistency and Inconsistency of system of Linear Simultaneous Equations using Matrix Inversion method
A system of linear equations can be represented in the form Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants. One way to solve a system of linear equations is by using the matrix inversion method. The matrix inversion method involves finding the inverse of the coefficient matrix and then multiplying it by the column vector of constants to get the column vector of variables.
Consistency and Inconsistency of a System of Linear Simultaneous Equations:
When solving a system of linear equations using the matrix inversion method, we can determine if the system is consistent or inconsistent by examining the determinant of the coefficient matrix. The determinant of the coefficient matrix is denoted by det(A).
If det(A) is not equal to zero, the system of equations has a unique solution and is said to be consistent. In this case, the inverse of the coefficient matrix exists, and we can use it to find the column vector of variables.
Example:
Consider the following system of equations:
2x + 3y = 8
4x + 5y = 14
This system of equations can be written in the form Ax = b, where:
A = [2 3; 4 5]
x = [x; y]
b = [8; 14]
The determinant of the coefficient matrix A is det(A) = (25) – (43) = 2. Since det(A) is not equal to zero, the system of equations has a unique solution and is said to be consistent.
We can find the inverse of the coefficient matrix A as follows:
A^{1} = (1/det(A)) * [5 3; 4 2]
A^{1} = (1/2) * [5 3; 4 2]
A^{1} = [5/2 3/2; 2 1]
We can now find the column vector of variables as follows:
x = A^{1} * b
x = [5/2 3/2; 2 1] * [8; 14]
x = [1; 3]
Therefore, the solution to the system of equations is x = 1 and y = 3.
If det(A) is equal to zero, the system of equations has either no solution or infinitely many solutions and is said to be inconsistent. In this case, the coefficient matrix does not have an inverse, and we cannot use the matrix inversion method to find the column vector of variables.
Example:
Consider the following system of equations:
2x + 3y = 8
4x + 6y = 16
This system of equations can be written in the form Ax = b, where:
A = [2 3; 4 6]
x = [x; y]
b = [8; 16]
The determinant of the coefficient matrix A is det(A) = (26) – (43) = 0. Since det(A) is equal to zero, the system of equations is inconsistent.
We cannot use the matrix inversion method to find the column vector of variables. In this case, we can examine the system of equations to determine if there are no solutions or infinitely many solutions. In this example, the second equation is a multiple of the first equation, so the system of equations has infinitely many solutions.
Describe the steps to calculate the solution of Linear Simultaneous Equations using the Rank method
A system of linear equations can be represented in the form Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants. The rank method is a way to solve a system of linear equations by using the rank of the augmented matrix.
Steps to calculate the solutions of Linear Simultaneous Equations using Rank Method:
The steps to solve a system of linear equations using the rank method are as follows:
Step 1: Write the augmented matrix
The augmented matrix is formed by appending the column vector of constants to the coefficient matrix. The augmented matrix is denoted by [A  b].
For example, consider the following system of linear equations:
2x + 3y = 8
4x + 5y = 14
The augmented matrix for this system is:
[2 3  8; 4 5  14]
Step 2: Determine the rank of the augmented matrix
The rank of the augmented matrix is the number of nonzero rows in its row echelon form. To find the row echelon form of the augmented matrix, we use elementary row operations, which include swapping two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row.
For example, we can reduce the augmented matrix [2 3  8; 4 5  14] to its row echelon form as follows:
[2 3  8; 0 1  6]
The row echelon form has two nonzero rows, so the rank of the augmented matrix is 2.
Step 3: Determine the rank of the coefficient matrix
The rank of the coefficient matrix A is the number of nonzero rows in its row echelon form. To find the row echelon form of the coefficient matrix, we use the same elementary row operations as in Step 2.
For example, we can reduce the coefficient matrix [2 3; 4 5] to its row echelon form as follows:
[2 3; 0 1/2]
The row echelon form has two nonzero rows, so the rank of the coefficient matrix A is 2.
Step 4: Determine the consistency of the system
If the rank of the augmented matrix is equal to the rank of the coefficient matrix, then the system of equations has a unique solution and is consistent. If the rank of the augmented matrix is greater than the rank of the coefficient matrix, then the system has infinitely many solutions and is consistent. If the rank of the augmented matrix is less than the rank of the coefficient matrix, then the system has no solution and is inconsistent.
For the example system of equations, the rank of the augmented matrix is 2, and the rank of the coefficient matrix is also 2. Therefore, the system of equations has a unique solution and is consistent.
Step 5: Find the column vector of variables
To find the column vector of variables, we can use back substitution. We start by solving for the last variable in the last row of the row echelon form, and then substitute this value into the secondtolast row, and so on.
For the example system of equations, the row echelon form is [2 3  8; 0 1  6]. Using back substitution, we can solve for y in the second row as y = 6. Then, we can substitute y = 6 into the first row to solve for x as x = 1.
Describe Eigenvalues and Eigenvectors and their properties
Eigenvalues and eigenvectors are fundamental concepts in linear algebra that have wide applications in various fields of science and engineering. They are used to study the properties of matrices and to solve systems of linear equations.
Let A be an n × n matrix. A nonzero vector x is called an eigenvector of A if there exists a scalar λ, called the eigenvalue, such that Ax = λx.
Properties of Eigenvalues and Eigenvectors
 Eigenvalues are scalars
Eigenvalues are scalars that may be real or complex numbers. They provide information about the properties of a matrix, such as whether it is invertible or singular.
 Eigenvectors are nonzero vectors
Eigenvectors are nonzero vectors that are associated with the eigenvalues. They are determined up to a scalar multiple and provide information about the direction of the linear transformation.
 Eigenvectors are linearly independent
If λ is an eigenvalue of A and x is its corresponding eigenvector, then any scalar multiple of x is also an eigenvector. The set of eigenvectors corresponding to distinct eigenvalues is linearly independent.
 Determining eigenvalues
Eigenvalues are obtained by solving the characteristic equation det(A – λI) = 0, where I is the identity matrix of the same size as A. The solutions of the characteristic equation are the eigenvalues of A.
For example, consider the following matrix A:
[2 1; 1 2]
The characteristic equation of A is:
det(A – λI) =
 2λ 1 
 1 2λ 
= (2λ)(2λ) – 1 = λ^{2} – 4λ + 3 = 0
Solving this quadratic equation, we get two eigenvalues: λ1 = 1 and λ2 = 3.
 Determining eigenvectors:
Once the eigenvalues of A are determined, the corresponding eigenvectors can be obtained by solving the equation (A – λI)x = 0, where x is the eigenvector associated with eigenvalue λ.
For the matrix A, we have two eigenvalues λ1 = 1 and λ2 = 3. For λ1 = 1, we have:
(A – λ1I)x =
 21 1 
 1 21 
=
 1 1 
 1 1 
Solving (A – λ1I)x = 0, we get the eigenvector corresponding to λ1 as x1 = [1; 1].
Similarly, for λ2 = 3, we have:
(A – λ2I)x =
 23 1 
 1 23 
=
 1 1 
 1 1 
Solving (A – λ2I)x = 0, we get the eigenvector corresponding to λ2 as x2 = [1; 1].
 Diagonalization:
A square matrix A can be diagonalized if it has n linearly independent eigenvectors. This means that we can find a diagonal matrix D and an invertible matrix P such that A = PDP^{1}, where D is a diagonal matrix whose entries are the eigenvalues of A and P is the matrix whose columns are the eigenvectors of A.
Describe Characteristic equation with example
In mathematics, a characteristic equation is a polynomial equation that is used to find the eigenvalues of a matrix. The eigenvalues of a matrix are the values of λ for which the equation Ax=λx has nonzero solutions. The characteristic equation is obtained by setting the determinant of (AλI) equal to zero, where I is the identity matrix of the same size as A.
For example, consider the following 2×2 matrix A:
A = [3 1]
[1 3]
To find the eigenvalues of A, we first form the matrix (AλI):
(AλI) = [3λ 1]
[1 3λ]
The determinant of this matrix is:
det(AλI) = (3λ)(3λ) – 1*1 = λ^{2} – 6λ + 8
Setting this determinant equal to zero gives the characteristic equation:
λ^{2} – 6λ + 8 = 0
Solving this quadratic equation gives the eigenvalues of A:
λ = 2 or λ = 4
Solving this quadratic equation gives the eigenvalues of A:
λ = 2 or λ = 4
So the characteristic equation is an equation whose roots give the eigenvalues of a matrix, and it is used in many applications of linear algebra, such as in solving systems of differential equations, computing the stability of dynamical systems, and in quantum mechanics.
Describe CayleyHamilton theorem and its application
The CayleyHamilton theorem is a fundamental result in linear algebra that states that every square matrix satisfies its own characteristic equation. In other words, if A is an n x n matrix, then its characteristic polynomial p(λ) satisfies the equation p(A) = 0, where 0 is the zero matrix.
Detailed Notes:
 The CayleyHamilton theorem is named after the mathematicians Arthur Cayley and William Hamilton, who independently discovered the result in the mid1800s.
 The theorem applies to any square matrix, whether it is diagonalizable or not.
The characteristic polynomial of an n x n matrix A is given by:
p(λ) = det(λI – A)
 where I is the n x n identity matrix, and det denotes the determinant.
The CayleyHamilton theorem states that if p(A) is evaluated for the matrix A, it gives the zero matrix:
 p(A) = det(AI – A^{2} – … + (1)^{n} A^{n}) = 0
 This means that every square matrix satisfies its own characteristic equation.
 The theorem has several applications in linear algebra and related fields, including matrix diagonalization, eigenvalue problems, and differential equations.
 One of the main applications of the CayleyHamilton theorem is in finding powers of a matrix.
 Suppose we want to compute the power A^{k} of a matrix A. One way to do this is to diagonalize A, so that it becomes a diagonal matrix D with eigenvalues on the diagonal. Then, we can compute D^{k} easily, by raising each diagonal element to the power k. Finally, we can transform the diagonal matrix D back to the original basis using the eigenvectors of A. However, diagonalization is not always possible, especially for nondiagonalizable matrices.
The CayleyHamilton theorem provides an alternative way to compute powers of a matrix, without diagonalization. Suppose we want to compute A^{k}. We can start by computing the characteristic polynomial p(λ) of A. Then, we substitute A for λ in the polynomial, and evaluate the resulting expression:
 p(A) = A^{n} + c_{{n1}}A^{{n1}} + … + c_{1}A + c_{0}I = 0
This gives us a linear combination of powers of A that equals the identity matrix I. Rearranging the terms, we can express any power of A in terms of lower powers of A:
A^{n} = c_{{n1}}A^{{n1}} – … – c_{1}A – c_{0}I
A^{{n+1}} = c_{{n1}}A^{{n}} – … – c_{1}A – c_{0}A
A^{{n+2}} = c_{{n1}}A^{{n+1}} – … – c_{1}A – c_{0}A^{2}
…
A^{k} = a_{{kn}}A^{{n}} + a_{{kn+1}}A^{{n+1}} + … + a_{{k1}}A^{{k1} }+ a_{kI}
where the coefficients a_{{kn}}, a_{{kn+1}}, …, a_{ k} can be computed recursively using the formula:
 A_{ {kn}} = c_{ {n1}}a_{ {kn1}} – … – c_{ 1}a_{ {k2}} – c_{ 0}a_{ {k1}}, for k >= n
Describe Vector Space with examples
A vector space is a collection of objects, called vectors, that can be added and scaled (multiplied) with each other in a consistent way. The objects and operations in a vector space satisfy certain axioms, which define the properties of a vector space. Vector spaces are a fundamental concept in linear algebra and have many applications in various fields of science and engineering.
 A vector space is a set V of objects, called vectors, together with two operations, vector addition and scalar multiplication, that satisfy the following axioms:
 Closure under addition: For any vectors u, v in V, their sum u + v is also in V.
 Associativity of addition: For any vectors u, v, w in V, (u + v) + w = u + (v + w).
 Commutativity of addition: For any vectors u, v in V, u + v = v + u.
 Identity element of addition: There exists a vector 0 in V, called the zero vector, such that u + 0 = u for any vector u in V.
 Inverse element of addition: For any vector u in V, there exists a vector u in V, called the additive inverse of u, such that u + (u) = 0.
 Closure under scalar multiplication: For any vector u in V and any scalar c, the product cu is also in V.
 Distributivity of scalar multiplication over vector addition: For any vectors u, v in V and any scalar c, c(u + v) = cu + cv.
 Distributivity of scalar multiplication over scalar addition: For any vector u in V and any scalars c, d, c(du) = (cd)u.
 Identity element of scalar multiplication: For any vector u in V, 1u = u, where 1 is the multiplicative identity of the underlying field.
 Some examples of vector spaces include:
 The set R^{n} of ndimensional column vectors, where addition and scalar multiplication are defined componentwise.
 The set of all polynomials of degree at most n over a field F, where addition and scalar multiplication are defined by adding and multiplying the coefficients of the polynomials.
 The set of all continuous functions on a closed interval [a, b], where addition and scalar multiplication are defined pointwise.
 The set of all matrices of size m x n over a field F, where addition and scalar multiplication are defined componentwise.
 Not all sets of objects and operations satisfy the axioms of a vector space. For example, the set of integers Z is closed under addition and multiplication, but it does not satisfy the axioms of a vector space, because it does not have a multiplicative identity.
 Vector spaces have many important properties and applications in various fields of science and engineering, such as physics, engineering, computer science, and economics. Some of the applications of vector spaces include:
 Linear transformations: A linear transformation is a function that maps one vector space to another in a way that preserves the vector space structure. Linear transformations are important in many areas of mathematics and science, such as geometry, physics, and computer graphics.
 Eigenvalues and eigenvectors: Eigenvalues and eigenvectors are important concepts in linear algebra that arise in the study of linear transformations and matrices. They have many applications in physics, engineering, and other fields.
Describe Subspace of Vector Space with examples
A subspace of a vector space is a subset of vectors that has two main properties: it is closed under vector addition and scalar multiplication. More specifically, a subspace of a vector space V is a nonempty subset U of V that satisfies the following three conditions:
 The zero vector 0 of V is in U.
 U is closed under vector addition: for any u, v in U, u + v is in U.
 U is closed under scalar multiplication: for any u in U and any scalar c, cu is in U.
In other words, a subspace of a vector space is a subset of vectors that is itself a vector space with respect to the same operations of vector addition and scalar multiplication as the original vector space.
Examples:
 The set of all vectors of the form (x, y, 0) in R^{3} is a subspace of R^{3.} This subset satisfies the three conditions for a subspace. It contains the zero vector (0, 0, 0), is closed under vector addition and scalar multiplication. For instance, if u = (x_{1}, y_{1}, 0) and v = (x_{2}, y_{2}, 0) are two vectors in this subspace, then their sum (u + v) is also of the form (x_{1 }+ x_{2}, y_{1} + y_{2}, 0) and hence is in the subspace. Similarly, if u is a vector in the subspace and c is a scalar, then the scalar multiple of u by c, cu = (cx, cy, 0), is also in the subspace.
 The set of all polynomials of degree at most n in the variable x is a subspace of the vector space P of all polynomials in x. This subset satisfies the three conditions for a subspace. It contains the zero polynomial, is closed under polynomial addition and scalar multiplication. For instance, if p(x) and q(x) are two polynomials of degree at most n, then their sum (p(x) + q(x)) is also of degree at most n and hence is in the subspace. Similarly, if p(x) is a polynomial of degree at most n and c is a scalar, then the scalar multiple of p(x) by c, cp(x), is also of degree at most n and hence is in the subspace.
Describe Linear dependence and Independence of vectors
In linear algebra, a set of vectors is said to be linearly dependent if there exists a nontrivial linear combination of these vectors that equals the zero vector. In contrast, a set of vectors is said to be linearly independent if no nontrivial linear combination of these vectors equals the zero vector.
More formally, let V be a vector space over a field F, and let v_{1}, v_{2}, …, v_{n} be vectors in V. Then, v_{1}, v_{2}, …, v_{n }are said to be linearly dependent if there exist scalars c_{1}, c_{2}, …, c_{n}, not all zero, such that:
c_{1}v_{1} + c_{2}v_{2} + … + c_{n}v_{n} = 0,
where 0 is the zero vector in V. On the other hand, v_{1}, v_{2}, …, v_{n} are said to be linearly independent if the only scalars c_{1}, c_{2}, …, c_{n} that satisfy the above equation are all zero.
Examples:
 Consider the set of vectors {v_{1}, v_{2,} v_{3}} in R^{3}, where v_{1} = (1, 2, 3), v_{2} = (2, 4, 6), and v_{3} = (3, 6, 9). We can see that these vectors are linearly dependent, because we can express v_{3} as a linear combination of v_{1} and v_{2}:
v_{3} = 3v_{1 }+ 0v_{2} – v_{3}.
This shows that we can obtain the zero vector by taking a nontrivial linear combination of these vectors, and hence they are linearly dependent.
 Consider the set of vectors {u_{1}, u_{2}, u_{3}} in R^{3}, where u_{1} = (1, 0, 0), u_{2} = (0, 1, 0), and u_{3} = (0, 0, 1). We can see that these vectors are linearly independent, because there are no nontrivial scalars that satisfy the equation:
c_{1}u_{1} + c_{2}u_{2} + c_{3}u_{3} = 0.
The only solution is c_{1} = c_{2} = c_{3} = 0, which means that these vectors cannot be expressed as a linear combination of each other. Therefore, they are linearly independent.
Describe Linear Transformation
Linear transformation is a mathematical concept that describes a function between two vector spaces that preserves the linearity of the original space. In other words, a linear transformation is a function that maps vectors from one vector space to another in a way that preserves their additive and scalar properties.
Mathematically, a linear transformation T from a vector space V to a vector space W is a function that satisfies the following properties:
 T(u + v) = T(u) + T(v) for all vectors u and v in V.
 T(kv) = kT(v) for all vectors v in V and scalars k.
These properties imply that a linear transformation preserves vector addition and scalar multiplication, which are the fundamental operations in vector spaces.
Linear transformations are commonly represented by matrices. In this case, the transformation of a vector v can be computed by multiplying the vector by a matrix A that represents the linear transformation:
T(v) = Av
Linear transformations have many applications in mathematics and science, including in physics, engineering, and computer graphics.
Verify the given Mappings on Linear or Nonlinear Transformation
Let’s consider the mapping T: R^{2} > R^{2} defined by T(x,y) = (3x + 2y, 4x – y).
To verify whether this mapping represents a linear or nonlinear transformation, we need to check whether it satisfies the properties of linearity. Specifically, we need to check whether T(u + v) = T(u) + T(v) and T(ku) = kT(u) for all vectors u, v in R^{2} and all scalars k.
Let’s start by checking the first property:
T(u + v) = T(x1 + x2, y1 + y2) = (3(x1 + x2) + 2(y1 + y2), 4(x1 + x2) – (y1 + y2))
= (3×1 + 2y1 + 3×2 + 2y2, 4×1 – y1 + 4×2 – y2)
T(u) + T(v) = T(x1, y1) + T(x2, y2) = (3×1 + 2y1, 4×1 – y1) + (3×2 + 2y2, 4×2 – y2)
= (3×1 + 2y1 + 3×2 + 2y2, 4×1 – y1 + 4×2 – y2)
Since T(u + v) = T(u) + T(v), the mapping T satisfies the first property of linearity.
Now let’s check the second property:
T(ku) = T(kx, ky) = (3kx + 2ky, 4kx – ky)
= k(3x + 2y, 4x – y) = kT(u)
Since T(ku) = kT(u), the mapping T satisfies the second property of linearity.
Therefore, we can conclude that the mapping T represents a linear transformation.
An example of how this mapping can be applied is as follows:
Let’s say we have a vector v = (1, 2) in R^{2}. Applying the linear transformation T to this vector, we get:
T(1, 2) = (3(1) + 2(2), 4(1) – 2) = (7, 2)
So, the vector (1, 2) is transformed into the vector (7, 2) under the linear transformation T.
Describe Matrix representation of a Linear Operation
The matrix representation of a linear operation is a way of representing a linear transformation using a matrix. This representation allows us to perform computations involving linear transformations using matrix algebra, which is often simpler and more convenient than working directly with the linear transformation.
Given a linear transformation T: V > W between two vector spaces V and W, we can represent T using a matrix A with respect to suitable bases of V and W. Let {v_{1}, v_{2}, …, v_{n}} and {w_{1}, w_{2}, …, wm} be bases of V and W, respectively. Then the matrix A representing T with respect to these bases is an m x n matrix whose ith column is the coordinate vector of T(vi) with respect to the basis {w_{1}, w_{2}, …, w_{m}} of W.
To compute the action of T on a vector v in V, we can simply multiply the matrix A by the column vector representing v with respect to the basis {v_{1}, v_{2}, …, v_{n}}. That is, if v = c_{1}v_{1} + c_{2}v_{2} + … + c_{n}v_{n}, then T(v) = Av, where v is a column vector and A is the matrix representing T with respect to the chosen bases.
Let’s consider an example to illustrate the matrix representation of a linear operation. Let T: R^{2} > R^{2} be the linear transformation defined by T(x,y) = (3x + 2y, 4x – y), as in the previous question. We can represent T using the standard basis {e_{1}, e_{2}} of R^{2} and the same basis {e_{1}, e_{2}} of R^{2}. Then the matrix A representing T with respect to these bases is given by:
A = [3 2]
[4 1]
To see how this matrix representation works, let’s compute the action of T on the vector (1, 2) using the matrix A:
[T(1, 2)]_{e} = A [1]
[2]
where [T(1, 2)]_{e} is the column vector representing T(1, 2) with respect to the standard basis {e_{1}, e_{2}}. Substituting the matrix A and the vector (1, 2), we get:
[T(1, 2)]_{e} = [7]
[2]
This tells us that T(1, 2) = 7e_{1} + 2e_{2}, which is consistent with the previous example. We could have also computed this directly using the definition of T, but the matrix representation provides a more efficient way to compute the action of T in general.
Describe Kernel and Image of a Linear Mapping
In linear algebra, the kernel and image of a linear mapping are two important concepts that describe the behavior of the mapping on vectors.
Let T: V > W be a linear mapping between vector spaces V and W. The kernel of T, denoted by ker(T), is the set of all vectors in V that are mapped to the zero vector in W. In other words, ker(T) = {v in V  T(v) = 0}, where 0 denotes the zero vector in W.
On the other hand, the image of T, denoted by im(T), is the set of all vectors in W that are obtained by applying T to some vector in V. In other words, im(T) = {w in W  w = T(v) for some v in V}.
To illustrate these concepts, let’s consider the linear mapping T: R^{3} > R^{2} defined by T(x, y, z) = (x – y + z, 2x + y – 3z). We can find the kernel and image of T as follows:
Kernel of T:
We need to find all vectors (x, y, z) in R^{3 }such that T(x, y, z) = (0, 0). That is, we need to solve the system of equations:
x – y + z = 0
2x + y – 3z = 0
Using standard methods of linear algebra, we can find that the solutions to this system are of the form (t, t, t) for some scalar t. Therefore, the kernel of T is the set:
ker(T) = {(t, t, t)  t in R}
Image of T:
To find the image of T, we need to determine all possible vectors in R^{2} that can be obtained by applying T to some vector in R^{3}. To do this, we can write any vector (a, b) in R^{2} as a linear combination of the columns of the matrix representing T with respect to the standard bases of R^{3 }and R^{2}. This gives:
(a, b) = a(1, 2) + b(1, 1) + 0(1, 3)
Therefore, any vector (a, b) in Rk. can be obtained as T(x, y, z) for some (x, y, z) in R^{3}, namely:
(x, y, z) = (a + b, 2a + b, a – 3b)
Hence, the image of T is the set:
im(T) = {(a, b) in R^{2}  a, b in R}
In other words, the image of T is the entire space R^{2}.
Thus, in this example, the kernel of T is a onedimensional subspace of R^{3}, while the image of T is the entire space R^{2}. These concepts play an important role in the study of linear transformations and have many applications in fields such as computer graphics, physics, and engineering.
Describe Rank and Nullity of a Linear Mapping and discuss the RankNullity Theorem
Let T: V > W be a linear mapping between finitedimensional vector spaces V and W. The rank of T, denoted by rank(T), is the dimension of the image of T, i.e., rank(T) = dim(im(T)). The nullity of T, denoted by nullity(T), is the dimension of the kernel of T, i.e., nullity(T) = dim(ker(T)).
The RankNullity Theorem states that for any linear mapping T: V > W, we have:
rank(T) + nullity(T) = dim(V)
In other words, the sum of the dimensions of the image and kernel of T equals the dimension of the domain of T.
To illustrate these concepts, let’s consider the linear mapping T: R^{3} > R^{2} defined by T(x, y, z) = (x – y + z, 2x + y – 3z). We can find the rank and nullity of T as follows:
Kernel and Image of T:
To find the kernel of T, we need to solve the equation T(x, y, z) = (0, 0) which leads to the system of homogeneous linear equations:
x – y + z = 0
2x + y – 3z = 0
We can solve this system using standard methods of linear algebra to obtain the general solution:
x = s – t
y = t
z = t
where s, t are arbitrary constants. Thus, the kernel of T is the set of all linear combinations of the vector (1, 0, 1) and (0, 1, 0), i.e.,
ker(T) = {(st, t, t)  s, t in R}
To find the image of T, we need to determine the set of all possible outputs of T. Since T is a linear mapping, we can express the output in terms of a linear combination of the columns of the matrix representing T:
T(x, y, z) = (x – y + z, 2x + y – 3z) = x(1, 2) + y(1, 1) + z(1, 3)
Thus, the image of T is the span of the columns of the matrix representing T, i.e.,
im(T) = {(a, b) in R^{2}  a, b in R}
Rank and Nullity of T:
Since the image of T is the entire space R^{2}, the rank of T is 2. To find the nullity of T, we need to determine the dimension of the kernel of T, which we found to be twodimensional. Hence, the nullity of T is 2. We can verify the RankNullity Theorem in this case:
rank(T) + nullity(T) = 2 + 2 = 4 = dim(R^{3}) = dim(V)
Therefore, the RankNullity Theorem holds for this example.
Calculate Kernel and Image of a Linear Transformation
Let’s consider the linear transformation T: R^{4} > R^{3} given by the matrix:
 1 2 3 4 
T = [A] =  0 1 1 1 
1 2 1 3 
To find the kernel and image of T, we need to perform row reduction on the augmented matrix [A  0], where 0 represents the zero vector in R^{3}. We obtain:
 1 2 3 4   0 
[R] =  0 1 1 1  *  0 
1 2 1 3   0 
Adding twice the first row to the second row and adding the first row to the third row, we obtain:
 1 2 3 4   0 
[R’] =  0 1 1 1  *  0 
 0 0 2 1   0 
Adding 3/2 times the third row to the first row and 2 times the third row to the second row, we obtain:
 1 2 0 1 
[R”] =  0 1 0 1 
 0 0 1 1 
Now, we can read off the kernel and image of T from the reduced row echelon form of [A  0]. To find the kernel of T, we solve the system of homogeneous linear equations represented by the reduced row echelon form:
x1 + 2×2 – x4 = 0
x2 + x4 = 0
x3 + x4 = 0
We can solve this system using standard methods of linear algebra to obtain the general solution:
x1 = s – 2t + u
x2 = t
x3 = u
x4 = t
where s, t, u are arbitrary constants. Thus, the kernel of T is the set of all linear combinations of the vector (1, 0, 0, 0), (2, 1, 0, 1), and (1, 0, 1, 0), i.e.,
ker(T) = {(s2t+u, t, u, t)  s, t, u in R}
To find the image of T, we look at the nonzero rows of the reduced row echelon form, which correspond to the pivot columns of the matrix A. We can see that the first, second, and third columns are pivot columns, so the corresponding columns of A form a basis for the image of T. Thus, the image of T is the span of the columns of the matrix:
 1 
[B] =  0 
1 
i.e.,
im(T) = {(a, b, c) in R^{3}  a – c = 0}
Therefore, the kernel of T is given by ker(T) = {(s2t+u, t, u, t)  s, t, u in R}, and the image of T is given by im(T) = {(a, b, c) in R^{3}  a – c = 0}.
Describe Rank of a Matrix
The rank of a matrix is a fundamental concept in linear algebra that measures the dimension of the column space of a matrix. In other words, the rank of a matrix is the maximum number of linearly independent columns in the matrix. The rank of a matrix can also be defined as the dimension of the row space of a matrix, which is the subspace spanned by the rows of the matrix.
Example:
Consider the matrix A =
 1 2 3 
 2 4 6 
 1 1 1 
To find the rank of matrix A, we can reduce it to row echelon form:
 1 2 3 
 0 0 0 
 0 0 0 
The row echelon form of the matrix shows that the first row of A is linearly independent of the other rows, and it corresponds to the pivot element in the matrix. Since there is only one pivot element in the matrix, the rank of A is 1.
Another way to find the rank of matrix A is to find the dimension of the column space of A. The column space of A is spanned by the columns of A. In this case, we can see that the first column of A is a linear combination of the other columns, so it is redundant. Thus, the dimension of the column space of A is 2. Since the rank of A is the same as the dimension of the column space of A, we can conclude that the rank of A is 2.
In summary, the rank of a matrix is an important concept in linear algebra that measures the dimension of the column space (or row space) of a matrix. It can be found by reducing the matrix to row echelon form and counting the number of pivot elements, or by finding the dimension of the column space (or row space) of the matrix.
Calculate the Rank of a Matrix using Echelon Form
The rank of a matrix can be calculated using row operations to reduce the matrix to row echelon form. Row echelon form is a special form of a matrix where the leading coefficient (i.e., the first nonzero element) of each row is to the right of the leading coefficient of the row above it. Once a matrix is in row echelon form, the rank of the matrix can be easily determined by counting the number of nonzero rows.
Example:
Consider the matrix A =
 1 2 3 
 2 4 6 
 1 1 1 
To find the rank of matrix A, we can use row operations to reduce the matrix to row echelon form:
 1 2 3 
 0 0 0 
 0 0 0 
The row echelon form of the matrix shows that the first row of A is linearly independent of the other rows, and it corresponds to the pivot element in the matrix. Since there is only one pivot element in the matrix, the rank of A is 1.
To reduce the matrix to row echelon form, we use the following row operations:
 Swap rows to bring a nonzero element to the top of the first column.
 Multiply a row by a nonzero scalar to make the leading coefficient of the row above it equal to 1.
 Subtract a multiple of one row from another row to eliminate the leading coefficient of the row below it.
Using these operations, we can perform the following steps to reduce matrix A to row echelon form:
 1 2 3 
 0 0 0 
 0 0 0 
Swap the first and second rows to bring a nonzero element to the top of the first column:
 2 4 6 
 1 2 3 
 1 1 1 
Subtract twice the first row from the second row to make the leading coefficient of the second row equal to 1:
 2 4 6 
 0 2 3 
 1 1 1 
Subtract the first row from the third row to eliminate the first nonzero element in the third row:
 2 4 6 
 0 2 3 
 0 3 5 
Multiply the second row by 1/2 to make the leading coefficient of the third row equal to 1:
 2 4 6 
 0 1 3/2 
 0 3 5 
Subtract 3 times the second row from the third row to eliminate the first nonzero element in the third row:
 2 4 6 
 0 1 3/2 
 0 0 11/2 
The resulting matrix is in row echelon form. Since there are only two nonzero rows, the rank of A is 2.
In summary, the rank of a matrix can be calculated by reducing the matrix to row echelon form using row operations and counting the number of nonzero rows. Row echelon form is a special form of a matrix where the leading coefficient of each row is to the right of the leading coefficient of the row above it.
Calculate the Rank of a Matrix using Normal Form
The rank of a matrix can also be calculated using its normal form. The normal form of a matrix is obtained by applying row and column operations to the matrix in order to reduce it to a diagonal form. The rank of the matrix is then equal to the number of nonzero diagonal entries.
Example:
Consider the matrix A =
 1 2 3 
 2 4 6 
 1 1 1 
To find the rank of matrix A, we can use row and column operations to reduce the matrix to normal form:
 1 0 0 
 0 0 0 
 0 0 0 
To reduce the matrix to normal form, we first use row operations to obtain a row echelon form:
 1 2 3 
 0 0 0 
 0 1 2 
Next, we apply column operations to obtain a diagonal form:
 1 0 0 
 0 0 0 
 0 0 0 
The resulting matrix is in normal form, and we can see that the rank of A is 1, since there is only one nonzero diagonal entry.
In summary, the rank of a matrix can be calculated using its normal form, which is obtained by applying row and column operations to the matrix in order to reduce it to a diagonal form. The rank is then equal to the number of nonzero diagonal entries.