Inverse of Matrix – Engineering Maths for GATE, ESE, AE/JE

The inverse of a matrix is a fundamental concept in Linear Algebra under Engineering Mathematics. It is widely used in solving linear equations and is frequently asked in GATE, ESE, AE/JE exams.

Basic Concept

The inverse of a square matrix A is denoted by A⁻¹ such that:

A · A⁻¹ = I

Where I is the identity matrix.

Condition for Inverse

• Matrix must be square
• Determinant ≠ 0

If |A| = 0 → Matrix is singular → inverse does NOT exist.

Inverse of 2×2 Matrix

If A =

A⁻¹ = (1 / (ad − bc)) ×

Inverse Using Adjoint Method

For higher order matrices:

A⁻¹ = Adj(A) / |A|

Where Adj(A) = transpose of cofactor matrix.

Important Properties

• (A⁻¹)⁻¹ = A
• (AB)⁻¹ = B⁻¹A⁻¹
• (A^T)⁻¹ = (A⁻¹)^T
• |A⁻¹| = 1 / |A|

Solved Examples

Example 1

Find inverse of:

|A| = (2×3 − 1×5) = 6 − 5 = 1

A⁻¹ =

Answer: Same matrix (since determinant = 1)

Example 2

Check if inverse exists:

|A| = (1×4 − 2×2) = 4 − 4 = 0

Answer: Inverse does NOT exist

Example 3

If |A| = 5, find |A⁻¹|

|A⁻¹| = 1 / 5

Shortcut Tricks

• Check determinant first
• Use formula directly for 2×2 matrix
• Use adjoint method for higher order
• Remember order reversal in (AB)⁻¹

Previous Year Concept Type Questions

Q1. When does inverse not exist?

Answer: When determinant = 0

Q2. If A is orthogonal, what is A⁻¹?

Answer: A^T

Exam Tips

• Always check determinant first
• Use properties to save time
• Avoid long adjoint calculations unless needed
• Practice PYQs

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