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Linear Algebra and its Applications
Found in: Page 395
Linear Algebra and its Applications

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

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Short Answer

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

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