Invertible Matrices: Definition, Theorems and Examples | Class 12 CBSE Maths

The final major topic in NCERT Chapter 3 is invertible matrices — square matrices that possess a multiplicative inverse. Understanding invertibility is essential not only for the Matrices chapter itself but also for Chapter 4 (Determinants), where you will learn a practical method to compute inverses using cofactors. This guide gives you the complete theoretical foundation needed for both.

Definition of Invertible Matrices

A square matrix A of order m is called invertible (or non-singular) if there exists another square matrix B of the same order m such that:

AB = BA = I

where I is the identity matrix of order m. When this condition holds, B is called the inverse of A and is denoted by A⁻¹.

Equivalently, A is the inverse of B, so A = B⁻¹.

A Concrete Example

Let A = [[2, 3], [1, 2]] and B = [[2, −3], [−1, 2]]. Are they inverses of each other?

Check AB:

AB = | 2×2 + 3×(−1)   2×(−3) + 3×2 |   =   | 4−3   −6+6 |   =   | 1  0 |   = I ✓
     | 1×2 + 2×(−1)   1×(−3) + 2×2 |        | 2−2   −3+4 |       | 0  1 |

Check BA:

BA = | 2×2 + (−3)×1   2×3 + (−3)×2 |   =   | 4−3   6−6 |   =   | 1  0 |   = I ✓
     |(−1)×2 + 2×1   (−1)×3 + 2×2  |        |−2+2  −3+4 |       | 0  1 |

Both AB = I and BA = I hold, so B = A⁻¹ (and A = B⁻¹).

Important Notes on Invertible Matrices

Only Square Matrices Can Be Invertible

A rectangular matrix does not possess an inverse. For products AB and BA to both be defined and equal, matrices A and B must be square matrices of the same order.

If B is the Inverse of A, then A is the Inverse of B

Invertibility is symmetric: A⁻¹ = B if and only if B⁻¹ = A.

Theorem 3: Uniqueness of the Inverse

Theorem: The inverse of a square matrix, if it exists, is unique.

Proof

Let A be a square matrix of order m, and suppose B and C are both inverses of A. We want to show B = C.

Since B is the inverse of A: AB = BA = I …(1) Since C is the inverse of A: AC = CA = I …(2)

Now:

B = BI = B(AC) = (BA)C = IC = C

Therefore B = C. The inverse is unique. ✓

This elegant proof uses only the associative law of matrix multiplication and the definition of the identity matrix.

Theorem 4: (AB)⁻¹ = B⁻¹A⁻¹ — The Reversal Law for Inverses

Theorem: If A and B are invertible matrices of the same order, then their product AB is also invertible, and:

(AB)⁻¹ = B⁻¹A⁻¹

Just like the reversal law for transpose, the inverse of a product reverses the order.

Proof

Starting from (AB)(AB)⁻¹ = I:

(AB)(AB)⁻¹ = I
A⁻¹(AB)(AB)⁻¹ = A⁻¹I          [pre-multiply by A⁻¹]
(A⁻¹A)B(AB)⁻¹ = A⁻¹           [associativity]
IB(AB)⁻¹ = A⁻¹                [A⁻¹A = I]
B(AB)⁻¹ = A⁻¹
B⁻¹B(AB)⁻¹ = B⁻¹A⁻¹           [pre-multiply by B⁻¹]
I(AB)⁻¹ = B⁻¹A⁻¹
(AB)⁻¹ = B⁻¹A⁻¹   ✓

This proof appears as a theorem in NCERT and the derivation itself is a standard 5-mark board question.

Matrices A and B are Inverses if and Only if AB = BA = I

The condition for two matrices to be inverses of each other is this two-way requirement: both AB = I and BA = I must hold. It is not sufficient for only one of these to be true.

This is directly tested as an MCQ:

Question: Matrices A and B are inverses of each other only if: (A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I Answer: D

Verifying that Two Matrices Are Inverses

To confirm that B = A⁻¹, you must compute both AB and BA and verify that each equals the identity matrix. If even one product differs from I, the matrices are not inverses.

Step-by-Step Verification

Given: A = [[3, 1], [5, 2]] and B = [[2, −1], [−5, 3]]. Verify B = A⁻¹.

Check AB:

AB = | 3×2 + 1×(−5)   3×(−1) + 1×3 |   =   | 6−5   −3+3 |   =   | 1  0 |   = I ✓
     | 5×2 + 2×(−5)   5×(−1) + 2×3 |        | 10−10  −5+6 |       | 0  1 |

Check BA:

BA = | 2×3 + (−1)×5   2×1 + (−1)×2 |   =   | 6−5   2−2 |   =   | 1  0 |   = I ✓
     |(−5)×3 + 3×5   (−5)×1 + 3×2  |        |−15+15 −5+6 |       | 0  1 |

Both products equal I, so B = A⁻¹ confirmed. ✓

This type of question — verify that two given matrices are inverses — is a reliable 3-mark problem in CBSE boards.

The Connection Between Invertibility and Linear Equations

When a system of linear equations is written in matrix form AX = B, the system has a unique solution X = A⁻¹B if and only if A is invertible. This connects Chapter 3 to Chapter 4 (Determinants), where matrices are applied to solve systems of linear equations — making invertibility one of the most consequential concepts in the entire unit.

Relationship to Determinants

A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0). This connection is fully developed in Chapter 4, where you will learn:

A⁻¹ = (1 / det(A)) × adj(A)

where adj(A) is the adjugate (adjoint) of A. Understanding invertibility now prepares you to compute inverses explicitly in the next chapter.

CBSE Board Exam Tips for Invertible Matrices

• Verify both ways: To confirm B = A⁻¹, always check both AB = I and BA = I.

• Uniqueness: If a matrix has an inverse, it has exactly one — you do not need to worry about multiple inverses.

• Reversal law: (AB)⁻¹ = B⁻¹A⁻¹ — the order reverses, exactly like (AB)’ = B’A’.

• No inverse for rectangles: Only square matrices of matching order can be inverses.

• MCQ alert: “AB = BA = I” is the definition of inverses — this exact statement appears as an MCQ option every year.

Suggested Reading on ChampionsPrep

• Matrix Multiplication: Non-Commutativity and Key Properties

• Transpose of a Matrix: Properties and the Reversal Law

• What is a Matrix? Introduction, Notation, and CBSE Exam Tips

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Complete Your Matrices Preparation

Invertible matrices complete the theoretical foundation of Chapter 3 and directly connect to Chapter 4 (Determinants), where you will compute inverses using cofactors. Master this chapter fully before moving on.

Practice every concept from Matrices with 100+ CBSE-aligned questions at ChampionsPrep . Start at just ₹10 — MCQs, long-form, and instant explanations.