Inverse of a Matrix - MathsTips.com

Given a matrix A, if there exists a matrix B such that AB = BA = I, then B is called inverse of A. When we multiply a number by its reciprocal we get 1 and when we multiply a matrix by its inverse we get Identity matrix. Inverse of A is denoted by $A^{-1}$ . The inverse is used to find the solution to a system of linear equation. Using determinant and adjoint, we can easily find the inverse of a square matrix.

The following results are extremely important:

Only a non-singular matrix can possess inverse i.e. a square matrix A possesses inverse if and only if determinant |A| $\neq$ 0.Then A is said to be invertible.
The inverse of a matrix, where exists, is unique i.e. a non-singular matrix A cannot possess different inverse, say B and C. If A is a non-singular matrix, then $A^{-1} = \frac{1}{|A|} adj A$

Algorithm to find inverse of a matrix:

Suppose a square matrix A is given whose inverse is to be obtained.

Find |A|. If |A| = 0, write “Inverse does not exist”. If |A| $\neq$ 0 write “Inverse exists” and proceed to step 2.
Find cofactor of all elements of A.
Write matrix of the cofactor of A.
Write adj A
$A^{-1} = \frac{1}{|A|} adj A$
Whether the inverse is correct verify it by $AA^{-1}$ = I (Identitiy Matrix).

Suppose a 2*2 matrix A whose determinant is not equal to 0. $A=\begin {bmatrix} a &b \\ c &d \end {bmatrix}$ where a,b,c,d are number, the inverse is $A^{-1}=\begin {bmatrix} a &b \\ c &d \end {bmatrix}^{-1} = \frac{1}{ad - bc} \begin {bmatrix} d &-b \\ -c &a \end {bmatrix}$

Example 1: Find the inverse of the following matrix : $B=\begin {bmatrix} 5 & -8\\ -10 &16 \end {bmatrix}$

Solution : $|B|=\begin {bmatrix} 5 & -8\\ -10 &16 \end {bmatrix}$ = 16*5 – (-10)(-8) = 80 -80 = 0.

$B^{-1}$ does not exist as |B| $\neq$ 0.

Example 2:

Find the inverse of matrix $A=\begin {bmatrix} 2 & -3\\ 1 &-2 \end {bmatrix}$

Solution: $|A|=\begin {bmatrix} 2 & -3\\ 1 &-2 \end {bmatrix}$ = -4 – (-3) = -1 |A| $\neq$ 0 and $A^{-1}$ exists.

The minor of the element are : $M_{11}$ = -2 $C_{11}$ = -2, $M_{12}$ = 1 $C_{12}$ = 1, $M_{21}$ = -3 $C_{21}$ = +3, $M_{22}$ = 2 $C_{22}$ = 2.

Cofactor matrix = $\begin {bmatrix} -2 & -1\\ 3 &2 \end {bmatrix}$ and Adj A = $\begin {bmatrix} -2 & 3\\ -1 &2 \end {bmatrix}$

$A^{-1} = \frac{1}{|A|} adj A$ = $= \frac{1}{-1}\begin {bmatrix} -2 & 3\\ -1 &2 \end {bmatrix}$ = $\begin {bmatrix} 2 & -3\\ 1 & -2 \end {bmatrix}$ .

Exercise

Find the inverse of the matrix $A=\begin {bmatrix} 1 &2 &-2 \\ -1&3&0 \\ 0&-2&1 \end {bmatrix}$ .
Find the inverse of matrix $F=\begin {bmatrix} 4 &2 &3 \\ 4&0&1 \\ 1&1&0 \end {bmatrix}$ .
Find the inverse of matrix $A=\begin {bmatrix} 1 &-4 \\ -2&3 \end {bmatrix}$ .
Find the inverse of matrix $D=\begin {bmatrix} 4 &-5 \\ 2&1 \end {bmatrix}$ .
Find the inverse of the matrix $G=\begin {bmatrix} 3 &-4 &1 \\ -3&6&-1 \\ 4&-6&2 \end {bmatrix}$ .

Algorithm to find inverse of a matrix:

Exercise

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