Determine which of the matrices in Exercises 7–12areorthogonal. If orthogonal, find the inverse.

11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)

A matrix\(P\) with, \(n \times n\) dimension, is orthogonal if it satisfies the equation\({P^T}P = {I_n}\).

It is given that\(P = \left( {\begin{aligned}{{}}{2/3}&{2/3}&{2/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\).

Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{2/3}&{2/3}&{2/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\left( {\begin{aligned}{{}}{2/3}&{2/3}&{2/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{\,\,\,29/9}&{ - 16/9}&{ - 8/9}\\{ - 16/9}&{\,\,\,21/9}&{\,\,\,8/9}\\{\, - 8/9}&{\,\,\,8/9}&1\end{aligned}} \right)\\ & \ne {I_3}\end{aligned}\)

As \({P^T}P \ne {I_3}\), it can be concluded that \(P\) is not an orthogonal matrix.