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 = I₃, then x + y equals
(a) 0
(b) −1
(c) 2
(d) none of these

Answer 1

(a) 0

$$\begin{gathered} \mathrm{Given}:AB=I_3 \\ \Rightarrow \left[\begin{array}{ccc}1& 2& x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& y\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& y+x\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore y+x=0 \end{gathered}$$