Question

Find the value of x for which the matrix product
$\left[\begin{array}{ccc}2& 0& 7\\ 0& 1& 0\\ 1& -2& 1\end{array}\right]\left[\begin{array}{ccc}-x& 14x& 7x\\ 0& 1& 0\\ x& -4x& -2x\end{array}\right]$ equal an identity matrix.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ \left[\begin{array}{ccc}2& 0& 7\\ 0& 1& 0\\ 1& -2& 1\end{array}\right]\left[\begin{array}{ccc}-x& 14x& 7x\\ 0& 1& 0\\ x& -4x& -2x\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}-2x+0+7x& 28x+0-28x& 14x+0-14x\\ 0+0+0& 0+1-0& 0+0-0\\ -x-0+x& 14x-2-4x& 7x-0-2x\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}5x& 0& 0\\ 0& 1& 0\\ 0& 10x-2& 5x\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore 5x=1 \\ \Rightarrow x=\frac{1}{5} \end{gathered}$$