Matrix equation, simplified from $4\times 3$ to $3\times 3$

Question

I am confused as to how this is simplified, The image shows the question and what the mark scheme says but I do not understand it.

Answer 1

Note that $$\left[ \begin{array}{c c} A & v \end{array} \right]\cdot \left[ \begin{array}{c} u \\ w \end{array} \right]=Au+wv=p\Leftrightarrow Au=p-wv,$$ where $$A=\left[ \begin{array}{c c c} 1 & 1 & 0\\ 1 & 2 & 1\\ 0 & 0 & 1 \end{array} \right],\quad v=\left[ \begin{array}{c} 0\\ 0\\ 1 \end{array} \right],\quad u=\left[ \begin{array}{c} x\\ y\\ z \end{array} \right], \quad p=\left[ \begin{array}{c} 1\\ 1\\ 0 \end{array} \right].$$