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
B
1
C
2
D
3

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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