The matrix  R(t) is defined by  R(t)=costsint−sintcost, then  R(s).R(t)=

The matrix  $R\left(t\right)$ is defined by  $R\left(t\right)=\left[\begin{array}{ll}\mathrm{cos}t& \mathrm{sin}t\\ -\mathrm{sin}t& \mathrm{cos}t\end{array}\right]$, then  $R\left(s\right).R\left(t\right)=$

1. A

$R\left(-s-t\right)$

2. B

$R\left(-s+t\right)$

3. C

$R\left(s-t\right)$

4. D

$R\left(s+t\right)$

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Solution:

$\begin{array}{l}R\left(s\right).R\left(t\right)=\left[\begin{array}{ll}\mathrm{cos}s& \mathrm{sin}s\\ -\mathrm{sin}s& \mathrm{cos}s\end{array}\right]\left[\begin{array}{ll}\mathrm{cos}t& \mathrm{sin}t\\ -\mathrm{sin}t& \mathrm{cos}t\end{array}\right]\\ =\left[\begin{array}{ll}\mathrm{cos}s.\mathrm{cos}t-\mathrm{sin}s.\mathrm{sin}t& \mathrm{cos}s.\mathrm{sin}t+\mathrm{sin}s.\mathrm{cos}t\\ -\mathrm{sin}s\mathrm{cos}t-\mathrm{sin}s\mathrm{cos}t& \mathrm{cos}s.\mathrm{cos}t-\mathrm{sin}s.\mathrm{sin}t\end{array}\right]\\ =\left[\begin{array}{ll}\mathrm{cos}\left(s+t\right)& \mathrm{sin}\left(s+t\right)\\ -\mathrm{sin}\left(s+t\right)& \mathrm{cos}\left(s+t\right)\end{array}\right]\\ =R\left(s+t\right)\end{array}$

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