Linear system/Cramer's rule/Fact/Proof

Proof

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

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

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

To fact