Question

$If A(α)\left[\begin{array}{ccc}Cos α&Sin α\\-Sin α&Cos α\end{array}\right],Prove A(α) A(-α)=I.$

Answer 1

HELLO DEAR,

$A(\alpha) = \left[\begin{array}{ccc}Cos\alpha&sin\alpha\\-sin \alpha& cos\alpha \end{array}\right]$

$A(\alpha).A(-\alpha) = \left[\begin{array}{cc}Cos\alpha&sin\alpha\\-sin \alpha& cos\alpha \end{array}\right]$ . $\left[\begin{array}{cc}Cos\alpha&-sin\alpha\\sin \alpha& cos\alpha \end{array}\right]$

= $\left[\begin{array}{cc}(cos\alpha.cos\alpha + sin\alpha.sin\alpha)&(-cos\alpha.sin\alpha + sin\alpha.cos\alpha)\\(-sin\alpha.cos\alpha + cos\alpha.sin\alpha)&(sin\alpha.sin\alpha + cos\alpha.cos\alpha)\end{array}\right]$

= $\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$ = I


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Dear Student,

Solution:

$If A(α)= \left[\begin{array}{ccc}Cos α&Sin α\\-Sin α&Cos α\end{array}\right]$

$A(-α) = \left[\begin{array}{ccc}Cos (-α)&Sin (-α)\\-Sin (-α)&Cos (-α)\end{array}\right]$

Since
$sin( - \alpha ) = - \sin( \alpha ) \\ \\ \cos( - \alpha ) = \cos( \alpha )$
$A(-α) = \left[\begin{array}{ccc}Cos (α)&-Sin (α)\\Sin (α)&Cos (α)\end{array}\right]$

A(α) A(-α) = $\left[\begin{array}{ccc}Cos α&Sin α\\-Sin α&Cos α\end{array}\right]$ +$\left[\begin{array}{ccc}Cos (α)&-Sin (α)\\Sin (α)&Cos (α)\end{array}\right]$

A(α) A(-α) =$\left[\begin{array}{ccc}cos^{2} \alpha+ sin^{2} \alpha&-sin \alpha \:cos \alpha+sin \alpha \:cos \alpha\\sin \alpha \:cos \alpha- sin \alpha \:cos \alpha & cos^{2} \alpha+ sin^{2} \alpha\end{array}\right]$

Since

${sin}^{2} x + {cos}^{2} x = 1$
A(α) A(-α) =$\left[\begin{array}{ccc} 1 & 0\\ 0& 1 \end{array}\right]$ = I

Hope it helps you