KC Sinha Mathematics Solution Class 12 Chapter 5 आव्यूह ( Matrices ) Exercise 5.2 (Q15-Q21)

Exercise 5.2

Ex-5.2

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Question 15

साबित करे कि दो आव्यूह (Prove that the product of two matrices)

$\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \cdot \sin \theta \\ \cos \theta \cdot \sin \theta & \sin ^{2} \theta\end{array}\right]$ तथा (and) $\left[\begin{array}{cc}\cos ^{2} \phi & \cos \phi \cdot \sin \phi \\ \cos \phi \cdot \sin \phi & \sin ^{2} \phi\end{array}\right]$ का गुणनफल एक शून्य आव्यूह है यदि θ और ɸ का अन्तप $\dfrac{\pi}{2}$ का विषम अपवर्त्य है । (is a zero matrix when θ and ɸ differ by an odd multiple of  $\dfrac{\pi}{2}$)

Sol :

$\theta-\phi=(2 n-1) \frac{\pi}{2} \quad \ldots n \in 2$

$\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta sin\theta \\ \cos \theta \sin \theta & si n^{2} \theta\end{array}\right]\left[\begin{array}{cc}\cos ^{2} \phi & \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin ^{2} \phi\end{array}\right]$

$=\left[\begin{array}{ccc}\cos ^{2} \theta \cos ^{2} \phi+\cos \theta \sin \theta & \cos ^{2} \theta \cos \phi \sin \phi+\cos \theta sin \theta \sin ^{2} \phi \\ \cos \theta \sin \theta \cos ^{2} \phi+\sin ^{2} \theta \cos \phi+sin \phi & \cos \theta \sin \theta \cos \phi \sin \phi+\sin ^{2} \theta \sin ^{2} \phi\end{array}\right]$

$\left[\begin{array}{ll}\cos \theta \cos \phi(\cos \theta \cos \phi+\sin \theta \sin \phi) & \cos \theta \sin \phi(\cos \theta \cos \phi+\sin \theta \sin \phi) \\ \sin \theta \cos \phi(\cos \theta \cos \phi+\sin \theta sin \phi) & \sin \theta \sin \phi\left(\cos \theta \cos \phi+\sin \theta sin \phi\right)\end{array}\right]$

$=\left[\begin{array}{ll}\cos \theta \cos \phi \cos (\theta-\phi) & \cos \theta \sin \phi \cos (\theta-\phi) \\ \sin \theta \cos \phi \cos (\theta-\phi) & \sin \theta \sin \phi \cos (\theta-\phi)\end{array}\right]$

$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=0$