Solve by the matrix method: $x + y + z = 6$, $2x - y + z = 6$, $x + 2y - z = 1$. What is $z$?

Explanation

1. 01

   $A = \begin{bmatrix}1 & 1 & 1 \\ 2 & -1 & 1 \\ 1 & 2 & -1\end{bmatrix}$ and $|A| = 7$, so $A^{-1}$ exists.

2. 02

   $\operatorname{adj} A = \begin{bmatrix}-1 & 3 & 2 \\ 3 & -2 & 1 \\ 5 & -1 & -3\end{bmatrix}$, so $X = \frac{1}{7}(\operatorname{adj} A)B = \begin{bmatrix}2 \\ 1 \\ 3\end{bmatrix}$.

3. 03

   So $x = 2$, $y = 1$ and $z = 3$. Check the third equation: $2 + 2 - 3 = 1$.