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?-B.

Short answer

The Matrix -B 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 that A andB are orthogonal matrices.

That is for any$\overrightarrow{v}\in {\mathrm{ℝ}}^n$it haslocalid="1659491109632" $\Vert B\overrightarrow{v}\Vert =\Vert \overrightarrow{v}\Vert$observed as,

localid="1659490317143" $$\begin{gathered} \Vert \left(-B\right)\overrightarrow{v}\Vert =\Vert -B\overrightarrow{v}\Vert \\ =\Vert -B\overrightarrow{v}\Vert ||-1|| \\ =||{\overrightarrow{v}}_1|| \end{gathered}$$

Thus, the matrix -B is an orthogonal matrix.