Question

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

Answer 1

$A^2=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$B^2=\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]=\left[\begin{array}{cc}{a}^2+b& a-1\\ ab-b& b+1\end{array}\right]$
$A^2+B^2=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]+\left[\begin{array}{cc}{a}^2+b& a-1\\ ab-b& b+1\end{array}\right]=\left[\begin{array}{cc}{a}^2+b-1& a-1\\ ab-b& b\end{array}\right]$
$A+B=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]+\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]=\left[\begin{array}{cc}a+1& 0\\ b+2& -2\end{array}\right]$
$(A+B{)}^2=\left[\begin{array}{cc}a+1& 0\\ b+2& -2\end{array}\right]+\left[\begin{array}{cc}a+1& 0\\ b+2& -2\end{array}\right]$
$=\left[\begin{array}{cc}(a+1{)}^2& 0\\ ab+2a-b-2& 4\end{array}\right]$
Given $(A+B{)}^2=A^2+B^2$
$\Rightarrow \left[\begin{array}{cc}(a+1{)}^2& 0\\ ab+2a-b-2& 4\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}(a^2+b+1)& a-1\\ ab-b& b\end{array}\right]$
$\Rightarrow a=1$ and $b=4$