---
title: "Inverse of product (AB)⁻¹"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:inverse-of-product-ab-inverse"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-of-product-ab-inverse
site: BCT Engineering
---

# Inverse of product (AB)⁻¹

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2072 Kartik, 5 marks; 2068 Baishakh, 5 marks (asked 2 times)
- Video: [Class 12th – Prove (AB) Inverse = B Inverse A Inverse | Matrices | Tutorials Point](https://www.youtube.com/watch?v=eGqWsVOP8mQ) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-of-product-ab-inverse
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

If A and B are two non-singular matrices, then prove that (AB)⁻¹ = B⁻¹A⁻¹.

## Short answer

Since (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AA⁻¹ = I and (B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹B = I, B⁻¹A⁻¹ is the inverse of AB. By uniqueness of inverse, (AB)⁻¹ = B⁻¹A⁻¹.

## Answer

Non-singular matrices (|A| ≠ 0, |B| ≠ 0) have inverses A⁻¹ and B⁻¹, with AA⁻¹ = A⁻¹A = I. The inverse of a matrix, when it exists, is unique.

#### Proof
First, AB is non-singular, since |AB| = |A||B| ≠ 0, so (AB)⁻¹ exists.

Multiply AB by B⁻¹A⁻¹ and use associativity:

```text
(AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹
             = A I A⁻¹
             = AA⁻¹
             = I
```

and in the other order:

```text
(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B
             = B⁻¹ I B
             = B⁻¹B
             = I
```

So B⁻¹A⁻¹ is a matrix whose product with AB, on either side, is I. By the definition and uniqueness of the inverse,

**(AB)⁻¹ = B⁻¹A⁻¹.** ∎

(This is the "reversal law" for inverses; it extends to any number of factors: (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why does (AB)^-1 exist?
   A. |AB| = |A||B| != 0
   B. A and B commute
   C. AB = I
   D. A = B
   Answer: A. Product of non-zero determinants is non-zero. (Key point: Existence)

2. The proof shows
   A. (AB)(A^-1 B^-1) = I
   B. (AB)(B^-1 A^-1) = I and (B^-1 A^-1)(AB) = I
   C. AB = BA
   D. (AB)(B^-1 A^-1) = AB
   Answer: B. Then uniqueness of the inverse gives the result. (Key point: Method)

3. In A(BB^-1)A^-1 what happens?
   A. BB^-1 = 0
   B. B cancels A
   C. BB^-1 = I, leaving AA^-1 = I
   D. A and B swap
   Answer: C. Associativity lets the inner pair cancel first. (Key point: Middle step)

## Related in Mathematics III

- [Product of orthogonal matrices](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aproduct-of-orthogonal-matrices) — Mathematics III, chapter 1, asked 4×
- [Product of orthogonal matrices](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aproduct-of-orthogonal-matrices-two) — Mathematics III, chapter 1, asked 4×
- [Transpose of a product of matrices](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Atranspose-of-a-product-of-matrices) — Mathematics III, chapter 1
- [Transpose of a product (AB)ᵀ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Atranspose-of-a-product-ab-t) — Mathematics III, chapter 1
- [Condition for matrix inverse to exist](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acondition-for-matrix-inverse-to-exist) — Mathematics III, chapter 1, asked 6×
- [Condition for matrix inverse to exist](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Acondition-for-matrix-inverse-to-exist-two) — Mathematics III, chapter 1, asked 6×

## Related in other subjects

- [Prove sampling theorem; Nyquist of product](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-nyquist) — Signals and Systems, chapter 4, asked 2×
- [Add two matrices](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Amatrix-add-two) — Computer Programming, chapter 7
- [Sum of two matrices in FORTRAN](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Afortran-matrix-sum) — Computer Programming, chapter 11
- [Cascaded two-ports: ABCD matrices multiply](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Acascade-transmission-matrix-product) — Electrical Circuit Theory, chapter 7