Studying graph theory I came across the proof of Kirchof's theorem for maximal trees ("The number of generating trees of a graph $G$ is equal to the determinant of the reduced Laplacian Matrix of $G$"), according to author Richard Stanley, it is necessary to apply a property of the determinant called multilinearity property, specifically: Given a matrix sum of the form (Eq1) $A_1 = A + E[i,i]$, its determinant can be expressed as: (Eq1) $\det(A_1) = \det(A) + \det(A')$ with $E[i, i]$ the matrix of the same dimensions of $A$ but with the entry $(i,i)$ as $1$, and all other entries equal to $0$, while $A'$ is the submatrix of $A$ with row and column $i$ eliminated. How to obtain these matrices is clear to me but my question is:
1) How is this property of multilinearity deduced?
2) What does it mean that the determinant is multilinear and how is this implied, specifically with (Eq1)?