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If F(x) = \( \begin{bmatrix} cos\,x&-sin\,x &0\\[0.3em] sin\,x&cos\,x &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\), show that F(x) . F(y) = F(x + y).

F(x) = [(cos x,-sin x,0)(sin x,cos x,0)(0,0,1)]

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Given : F(x) = \( \begin{bmatrix} cos\,x&-sin\,x &0\\[0.3em] sin\,x&cos\,x &0\\[0.3em] 0 & 0 & 1 \end{bmatrix}\).

To show : F(x) . F(y) = F(x + y).

Formula used :

If A is a matrix of order a x b and B is a matrix of order c x d , then matrix AB exists and is of order a x d ,

if and only if b = c

We know that,

cosx(cosy) – sinx (siny) 

= cos(x+y) and -cosx(siny) - sinx(cosy) 

= -sin(x+y)

F(x + y) = F(x) . F(y)

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