Question

If A = $\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]andB=\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]and$ ${AB=I}_3,thenx+yisequalto$

Answer 1

$A=\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

and $B=\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$AB=\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}1+0+0& -2+2+0& y+0+x\\ 0+0+0& 0+1+0& 0+0+0\\ 0+0+0& 0+0+0& 0+0+1\end{array}\right]=I_3$ where $I$ is the $3\times 3$ identity matrix.

$\Rightarrow \left[\begin{array}{ccc}1& 0& y+x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Comparing, we get

$y+x=0$

$\therefore x+y=0$