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Expert-verified Found in: Page 108 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # if ${{\mathbf{A}}}^{{\mathbf{2}}}{\mathbf{=}}{{\mathbf{I}}}_{{\mathbf{2}}}$ then matrix A must be either ${{\mathbf{I}}}_{{\mathbf{2}}}{\mathbf{}}{\mathbit{o}}{\mathbit{r}}{\mathbf{}}{\mathbf{-}}{{\mathbf{I}}}_{{\mathbf{2}}}$ .

The statement is false.

See the step by step solution

## Step 1: Explaining

Consider a matrix A satisfying the condition,

${A}^{2}={\mathrm{I}}_{2}$

Consider a matrix,

$\begin{array}{l}\mathrm{A}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\\ \ne {\mathrm{I}}_{2}\end{array}$

## Step 2: Result

Calculate ${A}^{2}$

${A}^{2}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}={I}_{2}$

Thus the given statement is false. ### Want to see more solutions like these? 