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 $AB=I,$ then $x+y$ equals

• A. $0$
• B. $-1$
• C. $2$
• D. none of these

Answer 1

The correct option is A $0$

$A=\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Also given, $AB=I$
$\left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}1& 0& x+y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow x+y=0$
Hence, option 'A' is correct.