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  • Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 11

Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 11

 

Answers (1)

Answer: A^{2}=\left[\begin{array}{cc}\cos 4 \theta & \sin 4 \theta \\ -\sin 4 \theta & \cos 4 \theta\end{array}\right]

Hint:

matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: A=\left[\begin{array}{cc}\cos 2 \theta & \sin 2 \theta \\ -\sin 2 \theta & \cos 2 \theta\end{array}\right]

Consider,

A^{2}=A A\\\\ A^{2}=\left[\begin{array}{cc}\cos 2 \theta & \sin 2 \theta \\ -\sin 2 \theta & \cos 2 \theta\end{array}\right]\left[\begin{array}{cc}\cos 2 \theta & \sin 2 \theta \\ -\sin 2 \theta & \cos 2 \theta\end{array}\right] \\\\ \\ =\left[\begin{array}{cc}(\cos 2 \theta) \times(\cos 2 \theta)+(\sin 2 \theta) \times(-\sin 2 \theta) & \cos 2 \theta \times \sin 2 \theta+\sin 2 \theta \times \cos 2 \theta \\ (-\sin 2 \theta) \times(\cos 2 \theta)+\cos 2 \theta \times(-\sin 2 \theta) & \sin 2 \theta \times(-\sin 2 \theta)+\cos 2 \theta \times \cos 2 \theta\end{array}\right]\\\\\\ =\left[\begin{array}{cc}\cos ^{2} 2 \theta-\sin ^{2} 2 \theta & \cos 2 \theta \sin 2 \theta+\sin 2 \theta \cos 2 \theta \\ -\cos 2 \theta \sin 2 \theta-\sin 2 \theta \cos 2 \theta & -\sin ^{2} 2 \theta+\cos ^{2} 2 \theta\end{array}\right]

We know that,  \cos ^{2} \theta-\sin ^{2} \theta=\cos ^{2} 2 \theta

 

A^{2}=\left[\begin{array}{cc} \cos (2 \times 2 \theta) & 2 \sin 2 \theta \cos 2 \theta \\ -2 \sin 2 \theta \cos 2 \theta & \cos (2 \times 2 \theta) \end{array}\right]

Again, we have 

\begin{array}{l} \\\\ \sin 2 \theta=2 \sin \theta \cos \theta\\\\ A^{2}=\left[\begin{array}{cc} \cos 4 \theta & \sin (2 \times 2 \theta) \\ -\sin (2 \times 2 \theta) & \cos 4 \theta \end{array}\right]\\\\ A^{2}=\left[\begin{array}{cc} \cos 4 \theta & \sin 4 \theta \\ -\sin 4 \theta & \cos 4 \theta \end{array}\right] \end{array}

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