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Q12E

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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.12. P = \left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)

The given matrix is an orthogonal matrix, and{P^{ - 1}} = \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\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$${P^T}P = {I_n}$$and its inverse is given as $${P^{ - 1}} = {P^T}$$.

It is given that P = \left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right).

Find the matrix$${P^T}P$$as shown below:

\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{aligned}} \right)\\ &= {I_4}\end{aligned}

## Step 2: Find the inverse

As $${P^T}P = {I_4}$$, 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}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\end{aligned}

Thus, the given matrix is an orthogonal matrix, and{P^{ - 1}} = \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right).