Cramer's rule

Statement

If we have a matrix $A \in {GL}_{n} (R)$ , and

A \vec{x} = \vec{b}

then the $i$ th entry of $\vec{x}$ , $x_{i}$ is given by

x_{i} = \frac{det A_{i}}{det A}

where $A_{i}$ is $A$ with the $i$ th column replaced by $\vec{b}$ .