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, Then Verify that at a = I2. - Mathematics

Sum

If\[A = \begin{bmatrix}\cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha\end{bmatrix}\] , then verify that AT A = I2.

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Solution

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

\[ A^T = \begin{bmatrix}\cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]

\[Now, \]

\[ A^T A = I_2 \]

\[Consider: LHS = A^T A\]

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

\[ = \begin{bmatrix}\cos^2 \alpha + \sin^2 \alpha & \cos \alpha \sin \alpha - \sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha - \cos \alpha \sin \alpha & \sin^2 \alpha + \cos^2 \alpha\end{bmatrix}\]

\[ = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = RHS\]

Hence proved.

Concept: Multiplication of Two Matrices
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 5 Algebra of Matrices
Exercise 5.4 | Q 8 | Page 55
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