Question

If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ then write $(aA+bB)(aA-bB)$ in the form of Matrix $A$.

Answer 1

Given:

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

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

$\therefore A=B$

Now,

$(aA+bB)(aA-bB)=(aA+bA)(aA-bA)$

$=(a+b)(a-b)\left(A\right)\left(A\right)=(a^2-b^2)\left(A^2\right)$

As $A$ is a unit matrix

$\therefore A^2=A$

Hence, the answer is $(a^2-b^2)A$