Cramer's Rule

Theorem
Let $n \in \N$.

Let $\mathbf b^T \in \R^n$.

Let $x_1, x_2, \ldots, x_n$ be real numbers.

Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_n}^T$.

Let $A$ be an invertible $n \times n$ matrix.

Let $A_i$ be the matrix obtained by replacing the $i$th column of $A$ with $\mathbf b$.

Let:


 * $A \mathbf x = \mathbf b$.

Then:


 * $x_i = \dfrac {\map \det {A_i} } {\map \det A}$

for $i \in \set {1, \ldots, n}$.