Inverse of 2×2 Matrix

Introduction

If A be a square matrix of order mxm then the inverse matrix of A is another matrix of the order mxm, denoted by A^-1 and defined as

$\text{AA}^{-1} = \text{A}^{-1}\text{A} = \text{I}_m$

where $\text{I}_m$ is a identity matrix of the same order as of the matrix A.

In order to find out the inverse matrix, we need to understand the following two terms:

• Singular Matrix: A square matrix is said to be a singular matrix if its determinant is equal to 0.
• Non-singular Matrix: A square matrix is said to be a non-singular matrix if its determinant is not equal to 0.

VERY IMPORTANT:

Only inverse of a non-singular square matrix can be calculated. A singular matrix cannot have its inverse matrix.

Inverse of a 2×2 Matrix

In general, two non-singular square matrices, let ‘A’ and ‘B’ are said to be inverse matrices if $\text{AB} = \text{BA} = \text{I}$, and all these matrices have the same order or size.

When A and B are inverse matrices, we can write:

$\text{A}^{-1} = \text{B}$ and $\text{B}^{-1} = \text{A}$

The inverse matrix of matrix A of following order and elements can be found as follows:

$\text{A} = \left [ \displaylines{ a & b \\ c & d} \right ]_{2×2}$

The determinant of matrix A is defined as:

$\text{|A|} = \left | \displaylines{ a & b \\ c & d } \right| = (ad – bc)$

The inverse of matrix A is defined as:

$\text{A}^{-1} = \dfrac{1}{\text{|A|}} \left ( \displaylines{ d & -b \\ -c & a} \right )$

where $\left ( \displaylines{d & -b \\ -c & a} \right )$ is said to be the adjoint matrix of the matrix A.

How to find the adjoint of a 2×2 matrix?

Let us consider a square matrix A as:

$\text{A} = \left [ \displaylines{ a & b \\ c & d} \right ]_{2×2}$

The adjoint matrix of A is written as:

$\left ( \displaylines{d & -b \\ -c & a} \right )$

In order to find the adjoint matrix, we need to interchange the elements of the principal diagonal without bringing any change to its signs.

Then, we need to multiply the elements of the secondary diagonal (or non-pricipal diagonal) with a – sign. Their positions should not be interchanged.

Solved Examples

Example 1: Find the inverse matrix of $\text{A} = \left ( \displaylines{3&-5\\-1&2} \right )$.

Solution:

Given,

A = $\left ( \displaylines{3&-5 \\-1&2} \right )$

First, we need to find the determinant to check whether the matrix is singular or non-singular matrix.

|A| = $\left | \displaylines{3&-5 \\-1&2} \right |$

$= 3(2) – (-5)(-1)$
$= 6 – 5$
$= 1$

Since, |A| ≠ 0, matrix A is a non-singular matrix and its inverse is possible.

Now,

Adjoint matrix of A = $\left ( \displaylines{ 2&5\\1&3} \right )$

$A ^{-1}= \frac{1}{|A|} (\text{Adjoint matrix of A})$

$= \frac{1}{1} \left ( \displaylines{2&5\\1&3} \right )$

$= \left ( \displaylines{2&5\\1&3} \right )$

Hence, the inverse of matrix A is found.