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}+\cdots +a_{1n}x_{n}&=&c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&c_{2}\\\vdots &\vdots &\vdots \\a_{n1}x_{1}+a_{n2}x_{2}+\cdots +a_{nn}x_{n}&=&c_{n}\end{matrix}}

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

Then the unique solution for

{}x_{j}

is given by

{}x_{j}={\frac {\det {\begin{pmatrix}a_{11}&\cdots &a_{1,j-1}&c_{1}&a_{1,j+1}&\cdots &a_{1n}\\\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &a_{n,j-1}&c_{n}&a_{n,j+1}&\cdots &a_{nn}\end{pmatrix}}}{\det M}}

.