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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

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

Answers (1)

Answer:

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.

Given:

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)

Since

[\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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