Question

For what values of $x:\left[121\right]\left[\begin{array}{ccc}1& 2& 0\\ 2& 0& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$

Answer 1

$\left[\begin{array}{ccc}1& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 0\\ 2& 0& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$

$\Rightarrow \left[\begin{array}{ccc}1+4+1& 2+0+0& 0+2+2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$

$\Rightarrow \left[\begin{array}{ccc}6& 2& 4\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$

$\Rightarrow 6\times 0+2\times 2+4x=0$

$\Rightarrow 4+4x=0$

$\Rightarrow x=-1$

Hence, the answer is $-1.$