If   A=cosθ−sinθsinθcosθ, then  A2 is

# If   $A=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$, then  ${A}^{2}$ is

1. A

$\left[\begin{array}{cc}{\mathrm{cos}}^{2}\theta & -{\mathrm{sin}}^{2}\theta \\ {\mathrm{sin}}^{2}\theta & {\mathrm{cos}}^{2}\theta \end{array}\right]$

2. B

$\left[\begin{array}{cc}\mathrm{cos}2\theta & -\mathrm{sin}2\theta \\ \mathrm{sin}2\theta & -\mathrm{cos}2\theta \end{array}\right]$

3. C

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

4. D

None of these

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### Solution:

$\begin{array}{l}A=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\\ {A}^{2}=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]×\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]\\ =\left[\begin{array}{cc}{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta & -\mathrm{cos}\theta \text{\hspace{0.17em}}\mathrm{sin}\theta -\mathrm{cos}\theta \mathrm{sin}\theta \\ \mathrm{sin}\theta \text{\hspace{0.17em}}\mathrm{cos}\theta +\mathrm{cos}\theta \text{\hspace{0.17em}}\mathrm{sin}\theta & -{\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta \end{array}\right]\\ =\left[\begin{array}{cc}\mathrm{cos}2\theta & -\mathrm{sin}2\theta \\ \mathrm{sin}2\theta & -\mathrm{cos}2\theta \end{array}\right]\end{array}$  +91

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