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#### Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 9

Answer: $A^{T}A=I_{2}$

Given: $A=\begin{bmatrix} \sin \alpha &\cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$

Hint: $I_{2}$ refers to an identity matrix with two rows and two columns.

Try to multiply$A^{T}$ with $A$.

Solution:

$A^{T}A=I_{2}$

Consider: $LHS=A^{T}A$

$=\begin{bmatrix} \sin \alpha &\cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}^{T}\begin{bmatrix} \sin \alpha &\cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$

$=\begin{bmatrix} \sin \alpha &-\cos \alpha \\ \cos \alpha & \sin \alpha \end{bmatrix}\begin{bmatrix} \sin \alpha &\cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$

$\begin{bmatrix} \sin ^{2 }\alpha +\cos^{2} \alpha &\sin \alpha \cos \alpha -sin \alpha \cos \alpha \\ \sin \alpha \cos\alpha -\sin \alpha \cos\alpha & \cos ^{2}\alpha +\sin^{2}\alpha \end{bmatrix}$

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

LHS=RHS

$A^{T}A=I_{2}$ is proved