Question

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

7. \(\left( {\begin{aligned}{{}{}}{.6}&{\,\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\)

Short answer

\(P\) is an orthogonal matrix and\({P^{ - 1}} = \left( {\begin{aligned}{{}{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).

Step-by-step solution

Find the characteristic equation

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

It is given that\(P = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\). Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0\\0&1\end{aligned}} \right)\\ &= {I_2}\end{aligned}\)

Find the inverse

As\({P^T}P = {I_2}\), it can be concluded that\(P\)is an orthogonal matrix. So, the inverse of matrix\(P\)is\({P^T}\). Find\({P^T}\), as follows:

\(\begin{aligned}{}{P^{ - 1}} &= {P^T}\\ &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\end{aligned}\)

Thus, \(P\) is an orthogonal matrix and \({P^{ - 1}} = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).