Question

If the$\mathbf{n}\mathbf{\times }\mathbf{n}$matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?AB.

Short answer

The Matrix AB is orthogonal.

Step-by-step solution

Definition of Orthogonal transformation

An$\mathbf{n}\mathbf{\times }\mathbf{n}$ matrix Ais orthogonal if and only if its columns form an orthonormal basis of${\mathbf{ℝ}}^{\mathbf{n}}$.

Verification whether the given matrix is orthogonal.

Given thatA andB are orthogonal matrices.

By theorem: The Product AB of two orthogonal$n\mathit{\times }n$ matricesA andB is orthogonal.

Therefore,AB is an orthogonal matrix.