$If AB = O, then for the matrices$ $A = [\begin{matrix} \cos^{2} θ & \cos θ \sin θ \\ \cos θ \sin θ & \sin^{2} θ \end{matrix}] and$ $B = [\begin{matrix} \cos^{2} ϕ & \cos ϕ \sin ϕ \\ \cos ϕ \sin ϕ & \sin^{2} ϕ \end{matrix}], θ - ϕ i s$

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$an odd multiple of π$

$an odd number of \frac{π}{2}$

$an even multiple of \frac{π}{2}$

answer is A.

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Detailed Solution

$\begin{matrix} We have \\ A B = [\begin{matrix} \cos^{2} θ & \cos θ \sin θ \\ \cos θ \sin θ & \sin^{2} θ \end{matrix}] [\begin{matrix} \cos^{2} ϕ & \cos ϕ \sin ϕ \\ \cos ϕ \sin ϕ & \sin^{2} ϕ \end{matrix}] \\ = \cos (θ - ϕ) [\begin{array}{l} \cos θ \cos ϕ & \cos θ \sin ϕ \\ \sin θ \cos ϕ & \sin θ \sin ϕ \end{array}] \\ Since A B = O \\ \Rightarrow c o s (θ - ϕ) = 0 \\ \Rightarrow (θ - ϕ) is odd multiples of \frac{π}{2} \end{matrix}$