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Q12E

Expert-verifiedFound in: Page 395

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

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:

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

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

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