Question

For what values of $x$.

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

Answer 1

We have:
$\left[\begin{array}{ccc}1& 2& 1\end{array}\right]\left[\begin{array}{c}1\\ 2\\ 1\end{array}\begin{array}{c}2\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$
$\Rightarrow \left[\begin{array}{ccc}1+4+1& 2+0+0& 0+2+2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$
$\Rightarrow \left[\begin{array}{ccc}6& 2& 4\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=0$
$\Rightarrow [6\left(0\right)+2\left(2\right)+4\left(x\right)]=0$
$\Rightarrow [4+4x]=\left[0\right]$
$\therefore 4+4x=0$

$\therefore x=-1$