If A = begin{bmatrix} cos^2 alpha & cos alpha | vedclass

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(C) Given matrices are $A = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix}$ and $B =

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an odd multiple of $\pi / 2$

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(C) Given matrices are $A = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \cos \beta \sin \beta \ \cos \beta \sin \beta & \sin^2 \beta \end{bmatrix}$.Calculating the product $AB$:$AB = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix} \begin{bmatrix} \cos^2 \beta & \cos \beta \sin \beta \ \cos \beta \sin \beta & \sin^2 \beta \end{bmatrix}$$= \begin{bmatrix} \cos^2 \alpha \cos^2 \beta + \cos \alpha \sin \alpha \cos \beta \sin \beta & \cos^2 \alpha \cos \beta \sin \beta + \cos \alpha \sin \alpha \sin^2 \beta \ \cos \alpha \sin \alpha \cos^2 \beta + \sin^2 \alpha \cos \beta \sin \beta & \cos \alpha \sin \alpha \cos \beta \sin \beta + \sin^2 \alpha \sin^2 \beta \end{bmatrix}$$= \begin{bmatrix} \cos \alpha \cos \beta (\cos \alpha \cos \beta + \sin \alpha \sin \beta) & \cos \alpha \sin \beta (\cos \alpha \cos \beta + \sin \alpha \sin \beta) \ \sin \alpha \cos \beta (\cos \alpha \cos \beta + \sin \alpha \sin \beta) & \sin \alpha \sin \beta (\cos \alpha \cos \beta + \sin \alpha \sin \beta) \end{bmatrix}$$= \cos(\alpha - \beta) \begin{bmatrix} \cos \alpha \cos \beta & \cos \alpha \sin \beta \ \sin \alpha \cos \beta & \sin \alpha \sin \beta \end{bmatrix}$.For $AB$ to be a null matrix,we must have $\cos(\alpha - \beta) = 0$.This implies $\alpha - \beta = (2n + 1) \frac{\pi}{2}$ for some integer $n$,which is an odd multiple of $\pi / 2$.

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