Mathematics → Subject Test → 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,