Question:

If $[1-1 x]\left[\begin{array}{rrr}0 & 1 & -1 \\ 2 & 1 & 3 \\ 1 & 1 & 1\end{array}\right]\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]=0$, find $x$

Solution:

Given : $\left[\begin{array}{lll}1 & -1 & x\end{array}\right]\left[\begin{array}{ccc}0 & 1 & -1 \\ 2 & 1 & 3 \\ 1 & 1 & 1\end{array}\right]\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]=0$

$\Rightarrow\left[\begin{array}{lll}0-2+x & 1-1+x & -1-3+x\end{array}\right]\left[\begin{array}{l}0 \\ 1 \\ 1\end{array}\right]=0$

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

$\Rightarrow[0+x-4+x]=0$

$\Rightarrow 2 x-4=0$

$\Rightarrow 2 x=4$

$\Rightarrow x=\frac{4}{2}$

$\therefore x=2$

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