Cramer's Rule

Definition

The solution of a Linear System Equation. It expresses the solution in terms of the determinants.

For the system of linear equations $\mathbf{Ax}=\mathbf{b}$, the elements of the column vector $\mathbf{x}$, $x_i$ are given by $x_i=\frac{\text{det}({\mathbf{A}}_i)}{\text{det}(\mathbf{A})}$ where ${\mathbf{A}}_i=[{\mathbf{a}}_1,{\mathbf{a}}_2,\dots ,{\mathbf{a}}_{i-1},\mathbf{b},{\mathbf{a}}_{i+1},\dots ]$ is the matrix formed by replacing $i$-th column of $\mathbf{A}$ by the column vector $\mathbf{b}$.