Cramer’s Rule

Consider $Ax=b$ $n$ linear equations in $n$ unknowns. $x=(x_1,\dots ,x_n{)}^t$.

If $\det (A)\ne 0$, $\exists !$ solution given by

$$x_k=\frac{\det (M_k)}{\det (A)},k=1,\dots ,n$$

where $M_k$ is obtained by replacing column $k$ of $A$ by $b$