<s>[INST] Cramer's Rule Question Use Cramer's rule to solve this system for z:
$$2x+y+z=1$$
$$3x+z=4$$
$$x-y-z=2$$
so my work is:
$$\frac{\left|\begin{matrix}
2 & 1 & 1\\
3 & 0 & 4\\
1 & -1 & 2
\end{matrix}\right|}{\left|\begin{matrix}
2 & 1 & 1\\
3 & 0 & 1\\
1 & -1 & 1
\end{matrix}\right|}$$
which gives $\frac{3}{-3}$ or $-1$. The solution is $1$, can someone tell me what I am doing wrong? 
 [/INST] The determinant in the denominator is incorrect. It should be 
$$\left|\begin{matrix}
2 & 1 & 1\\
3 & 0 & 1\\
1 & -1 & {\color{red} {-1}}
\end{matrix}\right|$$
which evaluates to (according to Sarrus' Rule):
$$(2\cdot0\cdot-1) + (1\cdot1\cdot1) + (1 \cdot 3 \cdot -1) - (1\cdot0\cdot1) - (1\cdot3\cdot-1) - (1\cdot2\cdot-1)\\
= 0 +1 - 3 - 0 +3 + 2  \\
= 3$$
</s>