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?3A.

Short answer

The Matrix 3A 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 oflocalid="1659490737513" ${\mathbf{ℝ}}^{\mathbf{n}}$.

Verification whether the given matrix is orthogonal.

Given that A and B are orthogonal matrices.

Let $A=[{\overrightarrow{v}}_1{\overrightarrow{v}}_2...{\overrightarrow{v}}_n]$where $i\ne j$.

Then,${\overrightarrow{V}}_i-\overrightarrow{V_j}=0$ and$\left|\left|\overrightarrow{v_i}\right|\right|=1$ for $i=1,2,...,n$.

Therefore, it gives $3A=\left[3{\overrightarrow{v}}_{1}3{\overrightarrow{v}}_1...3{\overrightarrow{v}}_n\right]$.

The columns are orthogonal but the length is not 1.

$$\begin{gathered} ||3{\overrightarrow{v}}_i||=3||{\overrightarrow{v}}_i|| \\ =3 \end{gathered}$$

Thus,3A is not an orthogonal matrix.