$A = \begin{bmatrix}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}$ and $B = \begin{bmatrix}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{bmatrix}$ satisfy $AB = 6I$. Use this to solve $x - y = 3$, $2x + 3y + 4z = 17$, $y + 2z = 7$. What is $z$?

Explanation

1. 01

   The system is $AX = C$ with $C = \begin{bmatrix} 3 \\ 17 \\ 7 \end{bmatrix}$, and $AB = 6I$ means $A^{-1} = \frac{1}{6}B$.

2. 02

   $X = \frac{1}{6}BC = \frac{1}{6}\begin{bmatrix}12 \\ -6 \\ 24\end{bmatrix} = \begin{bmatrix}2 \\ -1 \\ 4\end{bmatrix}$.

3. 03

   So $z = 4$.