If E (θ) = \(\begin{bmatrix} cos^{2}\theta &cos\theta sin\theta \\ cos\theta sin\theta& sin^{2}\theta \end{bmatrix}\) and θ and ɸ differ by and odd multiple of π / 2, then E (θ) E (ɸ) is a

1) unit matrix

2) null matrix

3) diagonal matrix

4) None of these

Solution: (2) null matrix

\(\begin{array}{l}E(\theta)=\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\\cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right] \\\therefore E(\emptyset)=\left[\begin{array}{cc}\cos ^{2} \emptyset & \cos \emptyset \sin \emptyset \\\cos \emptyset \sin \emptyset & \sin \emptyset\end{array}\right] \\\therefore E(\theta) \times E(\emptyset)=\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\\cos \theta \sin \theta & \sin ^{2} \theta\end{array}\right]\left[\begin{array}{cc}\cos ^{2} \emptyset & \cos \emptyset \sin \emptyset] \\\cos \emptyset \sin \emptyset & \sin \emptyset\end{array}\right] \\E(\theta) \times E(\emptyset)=\left[\begin{array}{cc}\cos \theta \sin \emptyset \cos (\theta-\emptyset) & \cos \theta \sin \emptyset \cos (\theta-\emptyset) \\\cos \theta \sin \emptyset \cos (\theta-\emptyset) & \cos \theta \sin \emptyset \cos (\theta-\emptyset)\end{array}\right] \\=E(\theta) \times E(\emptyset)=\left[\begin{array}{ll}0 & 0 \\0 & 0\end{array}\right]\left[\begin{array}{c}\because(\theta-\emptyset)=(2 n+1) \frac{\pi}{2} \\\cos (2 n+1) \frac{\pi}{2}=0\end{array}\right]\end{array}\)

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