Methods to Find Inverse of a Matrix

Methods to find the inverse of a matrix involve the inverse of a matrix formula and by elementary operations. The inverse of matrix A is represented as A^-1 which when multiplied by matrix A gives an identity matrix.

In this article, we will explore different methods to find the inverse of a matrix in detail along with the inverse of matrix definition and inverse of matrix properties.

What is Inverse of a Matrix?

Inverse of a Matrix is defined as the matrix when multiplied by the original matrix gives the identity matrix. If A is a matrix, then the inverse of matrix A is represented as A^-1. The inverse of a matrix can only be determined for a square matrix and the determinant is not equal to zero (i.e., non-singular matrix).

Inverse of a Matrix Definition

The matrix results in identify matrix when multiplied by the given matrix A then the matrix is called as Inverse matrix of A. It is denoted as A^-1.

AA^-1 = A^-1A = I

where,

• A^-1 is Inverse of Matrix A
• I is an Identity Matrix

Properties of Inverse of Matrix

Some properties of the Inverse of the Matrix are given below.

• (A^-1)^-1 = A
• (A^T)^-1 = (A^-1)^T
• (AB)^-1 = B^-1 A^-1
• (kA)^-1 = k^-1 A^-1 = (1/k)A^-1

Methods to Find Inverse of a Matrix

The different methods to find the inverse of a matrix are as follows.

• By Inverse of a Matrix Formula
• By Elementary Operations

Inverse of a Matrix by Inverse of Matrix Formula

The inverse of matrix formula is obtained by dividing adjoint of matrix by determinant of matrix. The adjoint of a matrix is the transpose of the cofactor matrix. The below formula represents the inverse of a matrix formula.

A^-1 = adj(A) / |A|

where,

• A^-1 is Inverse of Matrix A
• adj(A) is Adjoint of Matrix A
• |A| is Determinant of Matrix A

Steps to Find Inverse of Matrix by Inverse of Matrix Formula

The below steps are used to find inverse of matrix from the inverse matrix formula.

• Step 1: First, find the determinant of the matrix, if determinant is zero then, inverse of matrix does not exists and if the determinant is non-zero then follow further steps.

• Step 2: Find the adjoint matrix of the given matrix.

• Step 3: Then, divide the adjoint matrix by determinant of the matrix.

Inverse of Matrix by Elementary Transformations

The below steps are followed to find the inverse of a matrix using elementary transformations.

• Step 1: First, write the matrix as A = IA where I is identity matrix of same order as A.

• Step 2: Then perform row elementary operation or column elementary operation until we get identity matrix in LHS.

• Step 3: When identity matrix I is achieved by performing the row or column operation then we get I = BA

• Step 4: Then, B represents the inverse of matrix A.

Inverse of 2 × 2 Matrix

Inverse of 2 × 2 matrix A = [Tex]\begin {bmatrix} a & b \\ c & d \end{bmatrix}[/Tex] can be directly obtained by below formula.

A^-1 = [Tex]\bold{\frac{1}{ad – bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}[/Tex]

Examples of Methods to Find Inverse of a Matrix

Example 1: Find the inverse of the matrix P = [Tex]\begin {bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix}[/Tex] by Direct Method.

Solution:

We can find the inverse of matrix P by following formula

P^-1 = [Tex]\bold{\frac{1}{ad – bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}[/Tex]

P^-1 = [Tex]\frac{1}{(2\times1 )- (3\times 5)}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]

P^-1 = [Tex]\frac{1}{2- 15}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]

P^-1 = [Tex]\frac{1}{- 13}\begin{bmatrix} 1 & -3 \\ -5 & 2 \end{bmatrix}[/Tex]

Example 2: Find the inverse of matrix Q = [Tex]\begin {bmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{bmatrix}[/Tex] using inverse matrix formula.

Solution:

We can find the inverse of matrix Q by following formula.

Q^-1 = adj(Q) / |Q|

First, we find |Q|.

|Q| =[Tex] \begin {vmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{vmatrix} [/Tex]

|Q| = 1[Tex] \begin {vmatrix} 1 & 0\\ 2&3 \end{vmatrix} [/Tex] – 0 [Tex] \begin {vmatrix} 6 & 0\\ 5&3 \end{vmatrix} [/Tex]+ 4 [Tex] \begin {vmatrix} 6& 1 \\ 5&2 \end{vmatrix} [/Tex]

|Q| = 1[3 – 0] – 0 + 4[12 – 5]

|Q| = 1(3) + 4(7)

|Q| = 3 + 28

|Q| = 31

Now we will find adj(Q)

To find adj(Q) we find the cofactor matrix of Q as

adj Q = Transpose of cofactor matrix of Q

C_ij = (-1)^{i + j} Mij

where, M_ij is minor.

Cofactor(Q) = [Tex] \begin {bmatrix}(-1)^{1+1}\begin {vmatrix} 1 & 0\\ 2&3\\ \end{vmatrix} (-1)^{1+2}\begin {vmatrix} 6 & 0\\ 5&3\\ \end{vmatrix} (-1)^{1+3}\begin {vmatrix} 1 & 0\\ 2&3\\ \end{vmatrix} \\ \\ (-1)^{2+1}\begin {vmatrix} 0 & 4\\ 2&3\\ \end{vmatrix} (-1)^{2+2}\begin {vmatrix} 1 & 4\\ 5&3\\ \end{vmatrix} (-1)^{2+3}\begin {vmatrix} 1 & 0\\ 5&2\\ \end{vmatrix} \\ \\ (-1)^{3+1}\begin {vmatrix} 0 & 4\\ 1&0\\ \end{vmatrix} (-1)^{3+2}\begin {vmatrix} 1 & 4\\ 6&0\\ \end{vmatrix} (-1)^{3+3}\begin {vmatrix} 1 & 0\\ 6&1\\ \end{vmatrix} \\ \\ \end {bmatrix}[/Tex]

On solving above matrix we get

Cofactor of Q = [Tex]\begin {bmatrix} 3 & -18 & 7 \\ 8& -17 & -2\\ -4&24&1 \end{bmatrix} [/Tex]

adj(Q) = [Cofactor(Q)] ^T

adj(Q) = [Tex]\begin {bmatrix} 3 & -18 & 7 \\ 8& -17 & -2\\ -4&24&1 \end{bmatrix}^T[/Tex]

adj(Q) = [Tex]\begin {bmatrix} 3 & 8 & -4 \\ -1 8& -17 & 24\\ 7&-2&1 \end{bmatrix} [/Tex]

So, the inverse of matrix Q is given by

Q^-1 = (1 / 31) [Tex]\begin {bmatrix} 3 & 8 & -4 \\ -1 8& -17 & 24\\ 7&-2&1 \end{bmatrix} [/Tex]

Example 3: Find the inverse of matrix V = [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5 &0 & 2 \end{bmatrix}[/Tex] by elementary transformations.

Solution:

To find the inverse of the matrix V = [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] we will use row operation.

V = IV

[Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] [Tex]\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex]

R1 ← R1 – R3

[Tex]\begin {bmatrix} 1 & 2 & 1 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] V

R3 ← R3 – 5R1

[Tex]\begin {bmatrix} 1 & 2 & 1 \\ 0 & 1 & 4\\ 4&-10 &-3 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0\\ -5&0 & 6 \end{bmatrix}[/Tex] V

R1← R1 – 2R2

[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&-10 & -3 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5&0 & 6 \end{bmatrix}[/Tex] V

R3 ← R3 + 10R1

[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&0 & 37 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5&10 & 6 \end{bmatrix}[/Tex] V

R3 ← R3/37

[Tex]\begin {bmatrix} 1 & 0 & -7 \\ 0 & 1 & 4\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 1 & -2 & -1 \\ 0 & 1 & 0\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V

R1 ← R1 + 7R3

[Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 4\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 0 & 1 & 0\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V

R2 ← R2 – 4R3

[Tex]\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix}[/Tex] = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 9/37 & -3/37 & -24/37\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex] V

Since, the above expression is of form I = BV

Inverse of matrix V = [Tex]\begin {bmatrix} 2/37 & -4/37 & 5/37 \\ 9/37 & -3/37 & -24/37\\ -5/37&10/37 & 6/37 \end{bmatrix}[/Tex]

Practice Problems on Methods to Find Inverse of a Matrix

P1: Find the inverse of the matrix P = [Tex]\begin {bmatrix} 12 & 8 \\ 20 & 15 \end{bmatrix}[/Tex] by Direct Method.

P2: Find the inverse of matrix X = [Tex]\begin {bmatrix} 2 & 10 \\ 15 & 5 \end{bmatrix}[/Tex] by elementary transformations.

P3: Find the inverse of matrix B = [Tex]\begin {bmatrix} 3 & 1 & -1 \\ 2& -2 & 0\\ 1&2&-1 \end{bmatrix}[/Tex] by inverse matrix formula.

P4: Find the inverse of matrix D = [Tex]\begin {bmatrix} 2 & 3 & 1 \\ 1 & 1 & 2\\ 2&3&4 \end{bmatrix}[/Tex] by elementary transformations.

FAQs on Methods to Find Inverse of a Matrix

What is Inverse of a Matrix Formula?

Inverse of a matrix formula is given by:

A^-1 = adj(A) / |A|

How to Find an Inverse Matrix?

To find an inverse matrix we divide the adjoint matrix by determinant of matrix. Another way to find an inverse matrix is to perform elementary operations on it.

What are Different Ways to Find Inverse of a Matrix?

Different ways to find inverse of a matrix are:

• By Inverse of Matrix Formula
• By Elementary Transformations