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Found in: Page 395

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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)

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

See the step by step solution

## Step 1: 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}

## Step 2: 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).