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Question

If $$A = \begin{bmatrix}\cos x & \sin x\\ -\sin x & \cos x\end{bmatrix}$$ and $$A\ adj\ A = k\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$$ then the value of $$k$$ is


A
0
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B
1
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C
2
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D
3
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Solution

The correct option is A $$1$$
We know that,
$$A(adj\ A) = |A|I$$
Given, $$A = \begin{bmatrix}\cos x & \sin x\\ -\sin x & \cos x\end{bmatrix}$$
$$|A| = \cos^{2}x + \sin^{2}x = 1$$
$$|A| = \cos^{2} x + \sin^{2} x = 1$$
$$\therefore A(adj A) = 1\cdot I=kI$$
$$k = 1$$.  

Mathematics

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