---
title: "Transpose of a product (AB)ᵀ"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:transpose-of-a-product-ab-t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/transpose-of-a-product-ab-t
site: BCT Engineering
---

# Transpose of a product (AB)ᵀ

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2070 Chaitra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/transpose-of-a-product-ab-t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Prove that (AB)ᵀ = BᵀAᵀ where A is a matrix of size m×p and B is a matrix of size p×n.

## Short answer

(AB)ᵀ and BᵀAᵀ have same size n×m. Entry (j,i) of (AB)ᵀ is Σ aᵢₖ bₖⱼ; of BᵀAᵀ is Σ bₖⱼ aᵢₖ, equal by commutativity of scalar multiplication, so (AB)ᵀ = BᵀAᵀ.

## Answer

**Statement.** If A is of size m × p and B is of size p × n, then (AB)ᵀ = BᵀAᵀ.

#### Proof
**Orders.** AB has size m × n, so (AB)ᵀ has size n × m. Bᵀ is n × p and Aᵀ is p × m, so BᵀAᵀ is n × m. Both sides have the same order.

**Entries.** Let A = [aᵢₖ] and B = [bₖⱼ], 1 ≤ i ≤ m, 1 ≤ k ≤ p, 1 ≤ j ≤ n. By the definition of a product, (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ.

The (j, i) entry of (AB)ᵀ equals the (i, j) entry of AB:

```text
[(AB)ᵀ]ⱼᵢ = Σₖ aᵢₖ bₖⱼ        (k = 1 to p)
```

The (j, i) entry of BᵀAᵀ is, using (Bᵀ)ⱼₖ = bₖⱼ and (Aᵀ)ₖᵢ = aᵢₖ:

```text
[BᵀAᵀ]ⱼᵢ = Σₖ (Bᵀ)ⱼₖ (Aᵀ)ₖᵢ = Σₖ bₖⱼ aᵢₖ
```

Multiplication of numbers is commutative, so the two sums are equal for every j and i. Therefore (AB)ᵀ = BᵀAᵀ. ∎

**Note.** The order reverses: (ABC)ᵀ = CᵀBᵀAᵀ.

## Check yourself

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

1. A is m x p and B is p x n. The size of B^T A^T is:
   A. m x n
   B. p x p
   C. n x m
   D. n x p
   Answer: C. B^T is n x p and A^T is p x m, giving n x m, matching (AB)^T. (Key point: Orders)

2. (AB)_ij equals:
   A. a_ij b_ij
   B. Sum over k of a_ik b_kj
   C. Sum over k of a_ki b_jk
   D. a_ij + b_ij
   Answer: B. By the definition of matrix product. (Key point: Entry of product)

3. The (j,i) entry of (AB)^T is:
   A. The (j,i) entry of AB
   B. The (i,j) entry of AB
   C. Zero
   D. The (i,i) entry of AB
   Answer: B. Transposing swaps indices. (Key point: Transpose entry)

4. Why are Σ a_ik b_kj and Σ b_kj a_ik equal?
   A. Matrices commute
   B. Multiplication of numbers is commutative
   C. A and B are symmetric
   D. p = 1
   Answer: B. Only the numbers a_ik and b_kj are swapped in each product, not the matrices. (Key point: Equality of sums)

## Related in Mathematics III

- [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
- [Inverse of product (AB)⁻¹](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-of-product-ab-inverse) — Mathematics III, chapter 1, asked 2×
- [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×
- [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×

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