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

# Product of orthogonal matrices

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2078 Kartik, 5 marks
- Video: [Orthogonal Matrices and their Properties | Linear Algebra](https://www.youtube.com/watch?v=wGgg2ywmNIA) — Wrath of Math
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/product-of-orthogonal-matrices-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define an orthogonal matrix. Prove that the product of two orthogonal matrices of the same order is also orthogonal.

## Short answer

A square matrix Q is orthogonal if QQᵀ = I. If A and B are orthogonal, then (AB)(AB)ᵀ = A(BBᵀ)Aᵀ = AAᵀ = I, so their product AB is also orthogonal.

## Answer

#### Definition
A real square matrix Q is **orthogonal** if QQᵀ = QᵀQ = I, i.e. Qᵀ = Q⁻¹. (Its rows, and its columns, form an orthonormal set, and |Q| = ±1.) Example: the rotation matrix [cos θ −sin θ; sin θ cos θ].

#### Proof
Let A and B be orthogonal matrices of the same order n, so

```text
AAᵀ = AᵀA = I   and   BBᵀ = BᵀB = I
```

Use the reversal law (AB)ᵀ = BᵀAᵀ. Then

```text
(AB)(AB)ᵀ = (AB)(BᵀAᵀ)
          = A (BBᵀ) Aᵀ
          = A I Aᵀ
          = AAᵀ
          = I
```

Similarly,

```text
(AB)ᵀ(AB) = BᵀAᵀAB = Bᵀ(AᵀA)B = BᵀIB = BᵀB = I
```

Both products equal I, so AB is orthogonal. ∎

**Alternative (using inverses):** Orthogonality gives |A| = ±1 and |B| = ±1, so |AB| = ±1 ≠ 0, and (AB)⁻¹ = B⁻¹A⁻¹ = BᵀAᵀ = (AB)ᵀ.

**Example.** A = [0 1; −1 0] and B = [0 −1; −1 0] are both orthogonal, and AB = [−1 0; 0 1], which satisfies (AB)(AB)ᵀ = I.

## Check yourself

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

1. A real square matrix Q is orthogonal when:
   A. Q^2 = I
   B. Q = Q^T
   C. QQ^T = Q^TQ = I
   D. |Q| = 0
   Answer: C. Orthogonal means Q^T = Q^-1, i.e. QQ^T = Q^TQ = I. (Key point: Definition)

2. Which law is used for (AB)^T?
   A. (AB)^T = A^TB^T
   B. (AB)^T = B^TA^T
   C. (AB)^T = AB
   D. (AB)^T = -B^TA^T
   Answer: B. The transpose of a product reverses the order. (Key point: Reversal law)

3. In (AB)(AB)^T = A(BB^T)A^T, why does the middle simplify?
   A. BB^T = 0
   B. BB^T = A
   C. BB^T = I because B is orthogonal
   D. BB^T = B
   Answer: C. BB^T = I, leaving AA^T = I. (Key point: Key step)

## Close variants set in other exams

- [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×

## 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

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