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?A+B.

Short answer

The Matrix A+B is not 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 AandB are orthogonal matrices.

That is for any$\overrightarrow{v}\in {\mathrm{ℝ}}^n$it has$\Vert A\overrightarrow{v}\Vert =\Vert \overrightarrow{v}\Vert$ and $\Vert B\overrightarrow{v}\Vert =\Vert \overrightarrow{v}\Vert$.

$$\begin{gathered} \Vert \left(A+B\right)\overrightarrow{v}\Vert =\Vert -A\overrightarrow{v}+B\overrightarrow{v}\Vert \\ =\Vert A\overrightarrow{v}\Vert ||B\overrightarrow{v}|| \\ =||\overrightarrow{v}||+||\overrightarrow{v}|| \\ =2||\overrightarrow{v}|| \end{gathered}$$

As the length of each column vector is 2, A+B is not an orthogonal matrix.

Hence, the Matrix A+B is not orthogonal.