If $A=\left[\begin{array}{lll}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ and $A B=I_{3}$, then $x+y$ equals
(a) 0
(b) $-1$
(c) 2
(d) none of these
(a) 0
Given : $A B=I_{3}$
$\Rightarrow\left[\begin{array}{lll}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{ccc}1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
$\Rightarrow \quad\left[\begin{array}{ccc}1 & 0 & y+x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
The corresponding elements of two equal matrices are equal.
$\therefore y+x=0$
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