Cramer's rule
A short proof

A short proof of Cramer's rule can be given by noticing that x 1 {\displaystyle x_{1}} is the determinant of the matrix

On the other hand, assuming that our original matrix A is invertible, this matrix X 1 {\displaystyle X_{1}} has columns A − 1 b, A − 1 v 2, …, A − 1 v n {\displaystyle A^{-1}\mathbf {b},A^{-1}\mathbf {v} _{2},\ldots,A^{-1}\mathbf {v} _{n}}, where v n {\displaystyle \mathbf {v} _{n}} is the n -th column of the matrix A. Recall that the matrix A 1 {\displaystyle A_{1}} has columns b, v 2, …, v n {\displaystyle \mathbf {b},\mathbf {v} _{2},\ldots,\mathbf {v} _{n}}, and therefore X 1 = A − 1 A 1 {\displaystyle X_{1}=A^{-1}A_{1}}. Hence, by using that the determinant of the product of two matrices is the product of the determinants, we have

The proof for other x j {\displaystyle x_{j}} is similar.
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