Inverse of a Matrix

Do you know what a matrix is?

The collection of numbers that are arranged in rows and columns is known as a matrix.

For example, 1 4 -1 2 1 4 2 1 0   is a matrix.

No of rows = 3 and No. of columns = 3

Elements of matrix are denoted by aij,

where, i = number of row and j = no of column of that element

like, in the given example

a11 = 1, a23 = 4, a33 = 0 

Inverse Of Matrix-

For a non-singular square matrix A of order n, there exists a matrix of the same order A-1 such that,

A.A-1 = I

[Non-Singular Matrix = The matrix which has a non-zero determinant is known as non – a singular matrix.

Methods to calculate the inverse of matrix –

We have two methods to calculate the inverse of a matrix.

Method-1: By Elementary Row Transformation or Elementary Column Transformation

Method-2: By Matrix Formula

Method-1: By elementary Row Transformation or Elementary Transformation

By Elementary Row Transformation,

Here are steps by which you can find the inverse of a matrix using Elementary transformation,

Step – 1: Check whether the matrix is invertible or not, i.e. it is non-singular or not.

Step – 2: If A-1 exists, then start doing its inverse.

Step – 3: Write A = IA, I is the identity matrix of order same of matrix A

Step – 4: Now, our work is to make an LHS( Left Hand Side) identity matrix, which will be done by applying sequences of row transformations; the same transformations will also apply on RHS.

Step – 5: When you get I = BA stop doing transformations, B on RHS will be the inverse of matrix A-1

By elementary column transformation,

If you want to solve the inverse of a matrix by column transformation, then follow the above steps by doing column transformations in place of row transformation.

Example, Find inverse of following matrix using elementary transformation

A = 1 2 1 3

Step – 1: |A| = 3 – 2 = 1

A-1 will exists, as A is non- singular matrix of order 2.

Step – 2: By elementary row transformation,

A = I.A

1 2 1 3 = 1 0 0 1 . A

R2 R2  – R1

1 2 0 1 = 1 2 -1 1 . A

R1 R1 – 2R2

1 0 0 1 = 3 0 -1 1 .A

in side of RHS, we get an identity matrix so here we will stop our process, and we have got inverse of the matrix that is 3 0 -1 1 . 

You can also do this by elementary column transformation, you will get the same answer in both cases.

Method-2: By Matrix formula

Matrix inverse formula, is given by

A-1 = 1A. Adj(A)

Do you know how to find Adj (A) that is an Adjoint of A.

Well, Adjoint is the transpose of matrix [Aij]

where Aij is the cofactor of elements aij. 

For example, Find the inverse of matrix A = 1 2 1 0 2 3 1 0 -1

Ans. |A| = 2

Let’s find the adjoint of the matrix

To find the adjoint, firstly you have to find cofactors of the elements

A11 = -2 A12 = 3 A13 = -2

 A21 = 2   A22 = -2 A23 = 2

 A31 = 4 A32 = -3 A33 = 2

[Aij] = -2 3 -2 2 -2 2 4 -3 2

Adj(A) = -2 2 4 3 -2 -3 -2 2 2

A-1 = 1A. Adj(A)

A-1 = 12. -2 2 4 3 -2 -3 -2 2 2

This is the inverse of the matrix A

Properties of an Inverse of a Matrix-

Below is the list of the properties of the inverse of a matrix to be kept in mind for solving the problems of matrix-

Property 1 – The inverse of a matrix is unique. 

Property 2 – AB-1=B-1A-1 (Reversal law)

Property 3 – If A is an invertible square matrix; Then AT is also invertible an  AT-1= A-1T

Property 4 – The inverse of an invertible matrix is a symmetric matrix. 

Property 5 -A-1=A-1 

Property 6- A.A-1 = In = A-1.A

Property 7- (A-1)-1 = A 

Example: If a square matrix of order n, then the find value of: AAn

Sol: KAn×n = KnA

AAn=An ×A

⟹An+1

Example- What will be the Determinant of Inverse Matrix

  |A-1| = 1A

Proof:-

A × A-1 = I

det(A •A-1) = det(I)

or, det(A) × det(A-1) = det(I)

Since, det(I) = 1

det(A) × det(A-1) = 1

or, det(A-1) = 1 / det(A) 

Hence, proved.

Inverse Of Matrix-

For a non-singular square matrix A of order n, there exists a matrix of the same order A-1 such that,

A.A-1 = I

[Non-Singular Matrix = The matrix which has a non-zero determinant is known as non – a singular matrix.]

Methods to calculate the inverse of matrix –

We have two methods to calculate the inverse of a matrix.

Method-1: By Elementary Row Transformation or Elementary Column Transformation

Method-2: By Matrix Formula

Method-1: By elementary Row Transformation or Elementary Transformation

By Elementary Row Transformation,

Here are steps by which you can find the inverse of a matrix using Elementary transformation,

Step – 1: Check whether the matrix is invertible or not, i.e. it is non-singular or not.

Step – 2: If A-1 exists, then start doing its inverse.

Step – 3: Write A = IA, I is the identity matrix of order same of matrix A

Step – 4: Now, our work is to make an LHS( Left Hand Side) identity matrix, which will be done by applying sequences of row transformations; the same transformations will also apply on RHS.

Step – 5: When you get I = BA stop doing transformations, B on RHS will be the inverse of matrix A-1.

By elementary column transformation,

If you want to solve the inverse of a matrix by column transformation, then follow the above steps by doing column transformations in place of row transformation.

Example, Find inverse of following matrix using elementary transformation

A = 1 2 1 3

Step – 1: |A| = 3 – 2 = 1

A-1 will exists, as A is non- singular matrix of order 2.

Step – 2: By elementary row transformation,

A = I.A

1 2 1 3 = 1 0 0 1 . A

R2 R2  – R1

1 2 0 1 = 1 2 -1 1 . A

R1 R1 – 2R2

1 0 0 1 = 3 0 -1 1 .A

in side of RHS, we get an identity matrix so here we will stop our process, and we have got inverse of the matrix that is 3 0 -1 1 . 

You can also do this by elementary column transformation, you will get the same answer in both cases.

Method-2: By Matrix formula

Matrix inverse formula, is given by

A-1 = 1A. Adj(A)

Do you know how to find Adj (A) that is an Adjoint of A.

Well, Adjoint is the transpose of matrix [Aij]

where Aij is the cofactor of elements aij. 

For example, Find the inverse of matrix A = 1 2 1 0 2 3 1 0 -1

Sol. |A| = 2

Let’s find the adjoint of the matrix

To find the adjoint, firstly you have to find cofactors of the elements

A11 = -2 A12 = 3 A13 = -2

 A21 = 2   A22 = -2 A23 = 2

 A31 = 4 A32 = -3 A33 = 2

[Aij] = -2 3 -2 2 -2 2 4 -3 2

Adj(A) = -2 2 4 3 -2 -3 -2 2 2

A-1 = 1A. Adj(A)

A-1 = 12. -2 2 4 3 -2 -3 -2 2 2

This is the inverse of the matrix A

Properties of an Inverse of a Matrix-

Below is the list of the properties of the inverse of a matrix to be kept in mind for solving the problems of matrix-

Property 1 – The inverse of a matrix is unique. 

Property 2 – AB-1=B-1A-1 (Reversal law)

Property 3 – If A is an invertible square matrix; Then AT is also invertible an  AT-1= A-1T

Property 4 – The inverse of an invertible matrix is a symmetric matrix. 

Property 5 -A-1=A-1 

Property 6- A.A-1 = In = A-1.A

Property 7- (A-1)-1 = A 

Example: If a square matrix of order n, then the find value of: AAn

Sol: KAn×n = KnA

AAn=An ×A

⟹An+1

Example- What will be the Determinant of Inverse Matrix

  |A-1| = 1A

Proof:-

A × A-1 = I

det(A •A-1) = det(I)

or, det(A) × det(A-1) = det(I)

Since, det(I) = 1

det(A) × det(A-1) = 1

or, det(A-1) = 1 / det(A) 

Hence, proved.

Conclusion-

We have read about the inverse of a matrix. Also, we got some basic ideas about what the matrix is. We have even talked about the adjoint of a matrix and its cofactors.

Inverse Matrix is a very important tool to solve matrix equations. It helps us to manipulate these matrix equations into algebraic equations, so we can solve them easily. Above we have discussed the properties of the inverse matrix so that we can memorize it, which helps in easily solving the problems related to the matrix and inverse of a matrix. These all topics have been discussed with examples to make them understand.

The collection of numbers that are arranged in rows and columns is known as a matrix.

For example, 1 4 -1 2 1 4 2 1 0   is a matrix.

No of rows = 3 and No. of columns = 3

Elements of matrix are denoted by aij,

where, i = number of row and j = no of column of that element

like, in the given example

a11 = 1, a23 = 4, a33 = 0