Question

If $A=\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right],B=\left[\begin{array}{cc}2& a\\ -1& b\end{array}\right]$ and if ${\left(A+B\right)}^2=A^2+B^2$, find the values of $a$ and $b$.

Answer 1

$\begin{array}{c}A=\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right],B=\left[\begin{array}{cc}2& a\\ -1& b\end{array}\right]\\ \\ \left(A+B\right)=\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right]+\left[\begin{array}{cc}2& a\\ -1& b\end{array}\right]=\left[\begin{array}{cc}3& 2+a\\ -2& -2+b\end{array}\right]\\ \\ {\left(A+B\right)}^2=\left[\begin{array}{cc}3& 2+a\\ -2& -2+b\end{array}\right]\left[\begin{array}{cc}3& 2+a\\ -2& -2+b\end{array}\right]=\left[\begin{array}{cc}5-2a& 2+a+2b+ab\\ -2-2b& b^2-2a-4b\end{array}\right]\\ Now,\\ A^2=\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& -2\end{array}\right]=\left[\begin{array}{cc}-1& -2\\ 1& 2\end{array}\right]\\ \\ B^2=\left[\begin{array}{cc}2& a\\ -1& b\end{array}\right]\left[\begin{array}{cc}2& a\\ -1& b\end{array}\right]=\left[\begin{array}{cc}4-a& 2a+ab\\ 2-b& -a+b^2\end{array}\right]\\ \\ A^2+B^2=\left[\begin{array}{cc}-1& -2\\ 1& 2\end{array}\right]+\left[\begin{array}{cc}4-a& 2a+ab\\ 2-b& -a+b^2\end{array}\right]=\left[\begin{array}{cc}3-a& -2+2a+ab\\ -1-b& 2-a+b\end{array}\right]\\ so,\\ {\left(A+B\right)}^2=A^2+B^2\\ =\left[\begin{array}{cc}5-2a& 2+a+2b+ab\\ -2-2b& b^2-2a-4b\end{array}\right]=\left[\begin{array}{cc}3-a& -2+2a+ab\\ -1-b& 2-a+b\end{array}\right]\\ \\ 5-2a=3-a\\ a=2\\ -2-2b=-1-b\\ b=-1\end{array}$