User:Angegane/sandbox

Let $$D_j(\mathbf y)$$ be the determinant of the matrix $A$, where the $j$th column of $A$ is replaced by the vector $y$. Since the determinant is a linear function with respect to each column of $A$, $D_j$ in particular is linear with respect to $y$. It follows that


 * $$\begin{matrix}

D_j(\mathbf b) &=& D_j(A\mathbf x)\\ &=& D_j(\sum_{i=1}^n x_i\mathbf a_i)\\ &=& \sum_{i=1}^nx_iD_j(\mathbf a_i). \end{matrix}$$