User:Max i m 942/sandbox

A short proof of Cramer's rule can be given by noticing that x1 is the determinant of the matrix
 * $$X_1=\begin{bmatrix}

x_1 & 0 & 0 & \dots & 0\\ x_2 & 1 & 0 & \dots & 0\\ x_3 & 0 & 1 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ x_n & 0 & 0 & \dots & 1 \end{bmatrix}$$

On the other hand, this matrix has columns $$A^{-1}  b, A^{-1}  v_2 , \ldots, A^{-1}  v_n $$, where $$v_j$$ is the $$j$$ th column of the matrix $$A$$, so its determinant is $$\det A^{-1} \det A_1= \frac{\det A_1}{\det A}$$, as wanted.