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#### Need Solution for R.D.Sharma Maths Class 12 Chapter 4 Algebra of Matrices Exercise Multiple Choice Questions Question 30 Maths Textbook Solution.

Answer: The correct option is $\text { (A); } I \cos \theta+J \sin \theta$

Given:        \begin{aligned} &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ &J=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right] \\ &B=\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] \end{aligned}

Solution:

From the Matrices,

\begin{aligned} &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ &\Rightarrow I \cos \theta=\left[\begin{array}{cc} \cos \theta & 0 \\ 0 & \cos \theta \end{array}\right] \\ &\begin{aligned} J \sin \theta=\left[\begin{array}{cc} 0 & \sin \theta \\ -\sin \theta & 0 \end{array}\right] \mathbf{I} \end{aligned} \\ &\begin{array}{l} I \cos \theta+J \sin \theta=\left[\begin{array}{cc} \cos \theta & 0 \\ 0 & \cos \theta \end{array}\right]+\left[\begin{array}{cc} 0 & \sin \theta \\ -\sin \theta & 0 \end{array}\right] \\ =\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] \end{array} \\ &\Rightarrow B=I \cos \theta+J \sin \theta \end{aligned}

Thus, the correct option is (C).