Question

TRUE OR FALSE? If matrix is A orthogonal, then the matrix${\mathit{A}}^{\mathbf{2}}$ must be orthogonal as well.

Short answer

The given statement is true.

Step-by-step solution

Recollect the properties of the orthogonal a matrix.

According to the properties of a orthogonal matrix, a n x nmatrix Ais an orthogonal matrix, if $||A\overrightarrow{x}||\mathbf{=}||\overrightarrow{x}||$for all$\overrightarrow{\mathbf{x}}\mathbf{\in }{\mathit{R}}^{\mathbf{n}}$.

Check whether the given statement is a true or false.

The given statement is if the matrix A is orthogonal, then the matrix$A^2$ must be orthogonal as well.

Here, A is orthogonal. This means,$||A\overrightarrow{x}||=||\overrightarrow{x}||$ .

Then,

role="math" localid="1661256896384" $$\begin{gathered} ||A^2\overrightarrow{x}||=||A\left(A\overrightarrow{x}\right)|| \\ =||A\left(\overrightarrow{x}\right)|| \\ =||\left(\overrightarrow{x}\right)|| \end{gathered}$$

It is clear that, the given statement is true, by the properties of orthogonal matrices.