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

10. \(\left( {\begin{aligned}{{}}{1/3}&{\,\,2/3}&{\,\,2/3}\\{2/3}&{\,\,1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\)

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

It is given that\(P = \left( {\begin{aligned}{{}}{1/3}&{2/3}&{\,2/3}\\{2/3}&{1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\). Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P & = \left( {\begin{aligned}{{}}{1/3}&{2/3}&{\,2/3}\\{2/3}&{1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\left( {\begin{aligned}{{}}{1/3}&{2/3}&{\,2/3}\\{2/3}&{1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0&0\\0&1&0\\0&0&1\end{aligned}} \right)\\ &= {I_3}\end{aligned}\)

As\({P^T}P = {I_3}\), so it can be concluded that\(P\)is a orthogonal matrix. Also, the inverse of matrix\(P\)is\({P^T}\).

Find\({P^T}\)as follows:

\(\begin{aligned}{}{P^{ - 1}} &= {P^T}\\ &= \left( {\begin{aligned}{{}}{1/3}&{2/3}&{\,2/3}\\{2/3}&{1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\end{aligned}\)

Thus, \(P\) is an orthogonal matrix and\({P^{ - 1}} = \left( {\begin{aligned}{{}}{1/3}&{2/3}&{\,2/3}\\{2/3}&{1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\).