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

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

Answers (1)

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 multiplyA^{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

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