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

9. \(\left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\)

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

It is given that\(P = \left( {\begin{aligned}{{}}{ - 4/5}&{\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right)\). Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{ - 4/5}&{\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right)\left( {\begin{aligned}{{}}{ - 4/5}&{\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right)\\& = \left( {\begin{aligned}{{}}1&0\\0&1\end{aligned}} \right)\\ &= {I_2}\end{aligned}\)

As\({P^T}P = {I_2}\), so it can be concluded that\(P\)is an orthogonal matrix. Also, the inverse of matrix\(P\)is\({P^T}\).

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

\(\begin{aligned}{}{P^{ - 1}} &= {P^T}\\ &= \left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\end{aligned}\)

Thus, \(P\) is an orthogonal matrix and\({P^{ - 1}} = \left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\).