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Show that $\begin{array}{l} \cos \theta \cdot\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \\ {\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]=I} \end{array}$
Show that
$\begin{array}{l} \cos \theta \cdot\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \\ {\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]=I} \end{array}$
CBSE | Class 12 | Exercise 5F | Maths | Matrices | RS Aggarwal
Show that
$\begin{array}{l} \cos \theta \cdot\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \\ {\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]=I} \end{array}$

Solution:

We have $\cos \theta\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)+\sin \theta\left(\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right)=I$.
Use the addition rule and scalar multiplication of matrix as follow:
$\begin{array}{c}
\text { LHL }=\left(\begin{array}{cc}
\cos ^{2} \theta & \sin \theta \cos \theta \\
-\sin \theta \cos \theta & \cos ^{2} \theta
\end{array}\right) \\
+\left(\begin{array}{cc}
\sin ^{2} \theta & -\sin \theta \cos \theta \\
\sin \theta \cos \theta & \sin ^{2} \theta
\end{array}\right) \\
=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)
\end{array}$
$=I$
$=RHL$

i

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