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 = \begin {bmatrix} a & b \\ c & d \end{bmatrix} can be directly obtained by below formula.

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

Examples of Methods to Find Inverse of a Matrix

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

Solution:

We can find the inverse of matrix P by following formula

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

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

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

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

Example 2: Find the inverse of matrix Q = \begin {bmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{bmatrix} 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| = \begin {vmatrix} 1 & 0 & 4 \\ 6& 1 & 0\\ 5&2&3 \end{vmatrix}

|Q| = 1 \begin {vmatrix} 1 & 0\\ 2&3 \end{vmatrix} - 0 \begin {vmatrix} 6 & 0\\ 5&3 \end{vmatrix} + 4 \begin {vmatrix} 6& 1 \\ 5&2 \end{vmatrix}

|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) = \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}

On solving above matrix we get

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

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

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

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

So, the inverse of matrix Q is given by

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

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

Solution:

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

V = IV

\begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix} = \begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix} \begin {bmatrix} 6 & 2 & 3 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix}

R1 ← R1 - R3

\begin {bmatrix} 1 & 2 & 1 \\ 0 & 1 & 4\\ 5&0 & 2 \end{bmatrix} = \begin {bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0\\ 0&0 & 1 \end{bmatrix} V

R3 ← R3 - 5R1

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

R1← R1 - 2R2

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

R3 ← R3 + 10R1

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

R3 ← R3/37

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

R1 ← R1 + 7R3

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

R2 ← R2 - 4R3

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

Since, the above expression is of form I = BV

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

Practice Problems on Methods to Find Inverse of a Matrix

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

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

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

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