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Maths-

General

Easy

Question

If $straight A subscript alpha$ = $open square brackets table row cell cos invisible function application alpha end cell cell sin invisible function application alpha end cell row cell negative sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close square brackets$ , then which of following statement is true -

$A_{α}$ . $A_{β}$ = $A_{α + β}$ and( $A_{α}$ )n = $[\begin{matrix} \cos^{n} α & \sin^{n} α \\ - \sin^{n} α & \cos^{n} α \end{matrix}]$
$A_{α}$ $A_{β}$ = $A_{α + β}$ and( $A_{α}$ )n = $[\begin{matrix} \cos n α & \sin n α \\ - \sin n α & \cos n α \end{matrix}]$
$A_{α}$ . $A_{β}$ = $A_{α + β}$ and( $A_{α}$ )n = $[\begin{matrix} \cos^{n} α & \sin^{n} α \\ - \sin^{n} α & \cos^{n} α \end{matrix}]$
$A_{α}$ . $A_{β}$ = $A_{α + β}$ and( $A_{α}$ )n = $[\begin{matrix} \cos n α & \sin n α \\ - \sin n α & \cos n α \end{matrix}]$

The correct answer is: $straight A subscript alpha$ . $straight A subscript beta$ = $A subscript alpha plus beta end subscript$ and( $straight A subscript alpha$ )n = $open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets$

we are given matrix A and we are asked to find which of the given statements are true.
By using the properties of a matrix we verify that
$straight A subscript alpha$ . $straight A subscript beta$ = $A subscript alpha plus beta end subscript$ and( $straight A subscript alpha$ )n = $open square brackets table row cell cos invisible function application n alpha end cell cell sin invisible function application n alpha end cell row cell negative sin invisible function application n alpha end cell cell cos invisible function application n alpha end cell end table close square brackets$

Therefore the correct option is choice 4

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