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

# Transpose of a product of matrices

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2079 Bhadra, 1+4 marks
- Video: [Transpose of a matrix product | Matrix transformations | Linear Algebra | Khan Academy](https://www.youtube.com/watch?v=UdHgWK13x5I) — Khan Academy
- 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-of-matrices
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define the transpose of a matrix. Prove that the transpose of the product of two matrices is the product of their transposes taken in reverse order.

## Short answer

The transpose interchanges rows and columns. (AB)ᵀ entry equals (AB)Ɫ = Σ aᵢ bⱢ = Σ bⱢ aᵢ = (Bᵀ Aᵀ)Ɫ, so (AB)ᵀ = Bᵀ Aᵀ.

## Answer

The **transpose** of a matrix A of order m × n is the n × m matrix Aᵀ obtained by interchanging rows and columns; its (i, j) entry is aⱼᵢ.

#### Theorem
If A is m × n and B is n × p, then (AB)ᵀ = BᵀAᵀ.

#### Proof
**Orders.** AB is m × p, so (AB)ᵀ is p × m. Bᵀ is p × n and Aᵀ is n × m, so BᵀAᵀ is p × m. The orders agree.

**Entries.** Let A = [aᵢₖ] and B = [bₖⱼ]. The (j, i) entry of (AB)ᵀ is the (i, j) entry of AB:

```text
[(AB)ᵀ]ⱼᵢ = (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ
```

The (j, i) entry of BᵀAᵀ is

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

Since real or complex numbers commute under multiplication, the two sums are equal for every j and i. Hence (AB)ᵀ = BᵀAᵀ. ∎

**Extension:** (ABC)ᵀ = CᵀBᵀAᵀ, by applying the result twice.

**Example.** A = [1 2; 3 4], B = [0 1; 1 0]. AB = [2 1; 4 3], (AB)ᵀ = [2 4; 1 3]. Bᵀ Aᵀ = [0 1; 1 0][1 3; 2 4] = [2 4; 1 3] ✓.

## Check yourself

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

1. If A is m x n and B is n x p then B^T A^T is
   A. m x p
   B. n x n
   C. p x m
   D. p x n
   Answer: C. B^T is p x n and A^T is n x m. (Key point: Orders)

2. The (j,i) entry of (AB)^T is
   A. sum_k a_ik b_kj
   B. sum_k a_jk b_ki
   C. sum_k b_ik a_kj
   D. a_ij b_ji
   Answer: A. It is the (i,j) entry of AB. (Key point: Entry)

3. The sums are equal because
   A. A and B commute
   B. scalars commute under multiplication
   C. A is symmetric
   D. B is square
   Answer: B. b_kj a_ik = a_ik b_kj. (Key point: Why equal)

## Related in Mathematics III

- [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×
- [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
- [Hermitian and skew-Hermitian matrices](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahermitian-and-skew-hermitian-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×
- [Hermitian and skew-Hermitian matrices](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahermitian-and-skew-hermitian-matrices-two) — Mathematics III, chapter 1

## Related in other subjects

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- [Pass array; transpose 3×3 matrix](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Apass-array-transpose-3x3) — Computer Programming, chapter 7
- [Transpose of matrix using OOP](https://www.bctengineer.com/question.md?key=semester_3%2Fobject_oriented_programming%3Amatrix-transpose-oop) — Object Oriented Programming, chapter 1
- [Stiffness matrix: element matrices, member forces, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-element-matrices-600kn-bmd) — Theory of Structures II, chapter 3