#### Need solution for RD Sharma maths class 12 chapter 4 Algebra of Matrices exercise Fill in the blank question 15

$\begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix}$

Hint:

Use the basic method of multiplication.

Given:

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

Solution:

Calculate,

$A^{T}=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

So,

\begin{aligned} &A \times A^{T}=\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] \times\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \\ &A \times A^{T}=\left[\begin{array}{cc} \cos ^{2} \theta+\sin ^{2} \theta & -\cos \theta \sin \theta+\cos \theta \sin \theta \\ -\sin \theta \cos \theta+\cos \theta \sin \theta & \sin ^{2} \theta+\cos ^{2} \theta \end{array}\right] \\ &A \times A^{T}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}