Question

$If A=\left[\begin{array}{ccc}Cos α &sin α\\-sin α& cos α\end{array}\right], Prove AAᵀ=I. Deduce A⁻¹ =Aᵀ.$

Answer 1

HELLO DEAR,

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

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

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

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

I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Dear Student,

Solution:

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

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

A.A' = $\left[\begin{array}{ccc}Cos^{2} \alpha +Sin ^{2} \alpha&-Cos \alpha \:sin \alpha+Cos \alpha \:sin \alpha\\Cos \alpha \:sin \alpha-Cos \alpha \:sin \alpha& Cos^{2} [\alpha] +Sin ^{2} [\alpha \end{array}\right]$

As we know that

${sin}^{2} \alpha + {cos}^{2} \alpha = 1$

A.A'= $\left[\begin{array}{ccc}1 &0\\0&1 \end{array}\right] \:= I$

Hope it helps you