First slide

Operations of matrix

Question

If $A_{α} = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}]$ then $A_{α} A_{β}$ is equal to

Moderate

A

$A_{α β}$
B

$A_{α + β}$
C

$A_{α - β}$
D

None of these

Solution

$\begin{array}{l} We have A_{α} A_{β} = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}] [\begin{array}{cc} \cos β & \sin β \\ - \sin β & \cos β \end{array}] \\ = [\begin{array}{cc} \cos αcos β - \sin αsin β & \cos αsin β + \sin αcos β \\ - \sin αcos β - \cos αsin β & - \sin αsin β + \cos αcos β \end{array}] \\ = [\begin{array}{cc} \cos (α + β) & \sin (α + β) \\ - \sin (α + β) & \cos (α + β) \end{array}] = A_{α + β} \end{array}$

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