Question

If $3A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$, and $A{A}^{\prime }=I$, then $-x-y=$ ...................

Answer 1

$3A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$, and $A{A}^{\prime }=I$
$\Rightarrow \left(3A\right)\left(3A^{\prime }\right)=9I$
$\Rightarrow \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]\left[\begin{array}{ccc}1& 2& x\\ 2& 1& 2\\ 2& -2& y\end{array}\right]=\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]$
comparing terms with variables $x$ and $y$
$\Rightarrow x+2y=-4$ and $x-y=-1$
solving above equations gives $x=-2$ and $y=-1$
$\therefore -x-y=3$