Linear system/Cramer's rule/Fact/Proof

For an invertible matrix ${\displaystyle {}M}$, the solution of the linear system ${\displaystyle {}Mx=c}$ can be found by applying ${\displaystyle {}M^{-1}}$, that is, ${\displaystyle {}x=M^{-1}c}$. Using fact, this means ${\displaystyle {}x={\frac {1}{\det M}}(M^{\operatorname {adj} })c}$. For the ${\displaystyle {}j}$-th component, this means

${\displaystyle {}x_{j}={\frac {1}{\det M}}{\left(\sum _{k=1}^{n}(-1)^{k+j}(\det M_{kj})\cdot c_{k}\right)}\,.}$

The right-hand factor is the expansion of the determinant of the matrix shown in the numerator with respect to the ${\displaystyle {}j}$-th column.