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The matrix R(t) is defined by R(t)=[(cos...

The matrix R(t) is defined by `R(t)=[(cos t,sin t),(-sin t,cos t)]`. Show that `R(s)R(t)=R(s+t)`.

Text Solution

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`R(s)R(t)=[(cos s ,sin s),(cos t,sin t)][(cos t,sin t),(-sin t,cos t)]`
`=[(cos s cos t-sin s sin t,cos s sin t + sin s cos t),(-sin s cos t-cos s sin t,cos s cos t-sin s sin t)]`
`=[(cos(s+t),sin (s+t)),(-sin (s+t),cos (s+t))]`
`=R(s+t)`
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