Linear system/Cramer's rule/Fact

Cramer's rule

Let $K$ be a field, and let

\begin{matrix} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} & = & c_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} & = & c_{2} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} x_{1} + a_{n 2} x_{2} + \dots + a_{n n} x_{n} & = & c_{n} \end{matrix}

be an inhomogeneous linear system over $K$ . Suppose that the describing matrix $M = {(a_{i j})}_{i j}$ is invertible.

Then the unique solution for

x_{j}

is given by

x_{j} = \frac{det (\begin{matrix} a_{11} & \dots & a_{1, j - 1} & c_{1} & a_{1, j + 1} & \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n 1} & \dots & a_{n, j - 1} & c_{n} & a_{n, j + 1} & \dots & a_{n n} \end{matrix})}{det M}

.