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Q36E

Expert-verifiedFound in: Page 108

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 ${{\mathbf{I}}}_{{\mathbf{2}}}{\mathbf{}}{\mathit{o}}{\mathit{r}}{\mathbf{}}{\mathbf{-}}{{\mathbf{I}}}_{{\mathbf{2}}}$ .**

The statement is false.

Consider a matrix A satisfying the condition,

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

Consider a matrix,

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

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.

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