Question

If $A=\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $AB=I_3$, then find $x+y$.

Answer 1

Given $A=\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Here, $|A|=1,|B|=1$
Also given $AB=I_3$
$\Rightarrow A=B^{-1}$ ....(1)
Now, $adjB={\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ -y& 0& 1\end{array}\right]}^T=\left[\begin{array}{ccc}1& 2& -y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow B^{-1}=\left[\begin{array}{ccc}1& 2& -y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
So, equation (1) becomes
$\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 2& -y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\therefore x=-y\Rightarrow x+y=0$