Question

if matrix A [1 0 0 1] of order 2×2 and B is [0 1 -1 0] of order 2×2 show that (aA+bB) (aA-bB)=(a^2+b^2)A​

Answer 1

$\large\underline{\sf{Solution-}}$

Given matrices are

$\rm :\longmapsto\:A = \bigg[ \begin{matrix}1&0 \\ 0&1 \end{matrix} \bigg]$

and

$\rm :\longmapsto\:B = \bigg[ \begin{matrix}0&1 \\ - 1&0 \end{matrix} \bigg]$

So,

$\rm :\longmapsto\:aA + bB$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}a&0 \\ 0&a \end{matrix} \bigg] + \bigg[ \begin{matrix}0&b \\ - b&0 \end{matrix} \bigg]$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}a&b \\ - b&a \end{matrix} \bigg]$

Now,

$\rm :\longmapsto\:aA - bB$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}a&0 \\ 0&a \end{matrix} \bigg] - \bigg[ \begin{matrix}0&b \\ - b&0 \end{matrix} \bigg]$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}a& - b \\ b& a \end{matrix} \bigg]$

Now, Consider

$\rm :\longmapsto\:(aA + bB)(aA - bB)$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}a&b \\ - b&a \end{matrix} \bigg]\bigg[ \begin{matrix}a& - b \\ b& a \end{matrix} \bigg]$

$\rm \:  =  \:  \: \bigg[ \begin{matrix} {a}^{2} + {b}^{2} &0 \\ 0& {a}^{2} + {b}^{2} \end{matrix} \bigg]$

$\rm \:  =  \:  \: ( {a}^{2} + {b}^{2})\bigg[ \begin{matrix}1&0 \\ 0&1 \end{matrix} \bigg]$

$\rm \:  =  \:  \: ( {a}^{2} + {b}^{2})A$

Hence,

$\bf :\longmapsto\:(aA + bB)(aA - bB) = ( {a}^{2} + {b}^{2})A$