Why the determinant of a matrix with the sum of each row's elements equal 0 is 0? I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and it's true, but I couldn't proof it. Could you help me?

Let your matrix be called $A$.  Then set $x$ to be the column vector of all 1's.  You have $$Ax=0=0x$$
Hence $x$ is an eigenvector of $A$, while $0$ is an eigenvalue. Since the determinant is the product of the eigenvalues, and one of those is zero, the determinant must be zero.
