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Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 54 sub question (i) math

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Hence proved

P(x) P(y)=P(x+y)=P(y) P(x)

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.


P(x)=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]

We have,

\\\\ P(x) P(y)=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos y & \sin y \\ -\sin y & \cos y\end{array}\right] \\\\\\ P(x) P(y)=\left[\begin{array}{cc}\cos x \cos y-\sin x \sin y & \sin y \cos x+\sin x \cos y \\ -\sin x \cos y-\cos x \sin y & -\sin x \sin y+\cos x \cos y\end{array}\right]

P(x) P(y)=\left[\begin{array}{cc} \cos (x+y) & \sin (x+y) \\ -\sin (x+y) & \cos (x+y) \end{array}\right]=P(x+y)


[\cos x \cos y-\sin x \sin y=\cos (x+y), \sin y \cos x+\sin x \cos y \\\\ =\sin (x+y),-\sin x \cos y- \cos x \sin y=-\sin (x+y)]

We have,

\\\\ P(y) P(x)=\left[\begin{array}{cc} \cos (x+y) & \sin (x+y) \\ -\sin (x+y) & \cos (x+y) \end{array}\right]=P(x+y)\\\\ P(x) P(y)=P(x+y)=P(y) P(x)

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