Question

If [x 4 1]$\left[\begin{array}{ccc}2& 1& 2\\ 1& 0& 2\\ 0& 2& -4\end{array}\right]\left[\begin{array}{c}x\\ 4\\ -1\end{array}\right]$ = 0, find x.

Answer 1

$$\begin{gathered} \mathrm{Given}:\left[\begin{array}{ccc}x& 4& 1\end{array}\right]\left[\begin{array}{ccc}2& 1& 2\\ 1& 0& 2\\ 0& 2& -4\end{array}\right]\left[\begin{array}{c}x\\ 4\\ -1\end{array}\right]=0 \\ \Rightarrow \left[\begin{array}{ccc}2x+4+0& x+0+2& 2x+8-4\end{array}\right]\left[\begin{array}{c}x\\ 4\\ -1\end{array}\right]=0 \\ \Rightarrow \left[\begin{array}{ccc}2x+4& x+2& 2x+4\end{array}\right]\left[\begin{array}{c}x\\ 4\\ -1\end{array}\right]=0 \\ \Rightarrow \left[\begin{array}{c}2x^2+4x+4x+8-2x-4\end{array}\right]=0 \\ \Rightarrow 2x^2+6x+4=0 \\ \Rightarrow x^2+3x+2=0 \\ \Rightarrow x^2+2x+x+2=0 \\ \Rightarrow x\left(x+2\right)+1\left(x+2\right)=0 \\ \Rightarrow \left(x+2\right)\left(x+1\right)=0 \\ \Rightarrow x+2=0\mathrm{or}x+1=0 \\ \Rightarrow x=-2\mathrm{or}x=-1 \\ \therefore x=-2,-1 \end{gathered}$$