Inverse of a Matrix

How to find Inverse of a Matrix?
To find inverse of a matrix, first we must check whether the matrix is singular or invertible. Here we find inverse of a 2 x 2 matrix.

Singular OR Invertible Matrix

Consider a Matrix

If ad - bc = 0, then the matrix is said to be a Singular Matrix.
If ad - bc 0, then the matrix is said to be an Invertible Matrix.
The inverse of a matrix only exists if the matrix is invertible.

Theorem

Consider a matrix

If the matrix is invertible, then its inverse is,

Practice Problems

Example 1. Show that A is an invertible matrix and find inverse, when

Solution:
First check whether the matrix is invertible or not

ad - bc = (2)(4) - (2)(-3) = 8 + 6 = 14 0

So, the matrix is invertible, now find its inverse,


Example 2. Find inverse of the matrix

Solution:
First check whether the matrix is invertible or not

ad - bc = (4)(-5) - (2)(-10) = -20 + 20 = 0

So, the matrix is singular. And its inverse does not exists.

Consider question No.3 of GRE Mathematics (GR9768)

Example 3. If

is invertible under matrix multiplication, then its inverse is?
Solution:
Recall the definition (theorem) of inverse of a matrix
If

is invertible, then

So the inverse of the given matrix is,