Here is a related post that I find interesting.
Theorem (Minkowski’s Criterion). Let be an matrix with real entries such that the diagonal entries are all positive, off diagonal entries are all negative, and the row sums are all positive. Then .
This is a nice criterion and is not very difficult to prove, but for a random matrix it is asking too much. To decide whether a matrix is singular one usually looks for a row/column consisting of zeros or adding up to zero. The following result gives sufficient conditions for this to work. Unfortunately, it does not generalise Minkowski’s result.
Theorem. Let be a matrix with real entries such that its row sums are all , its lower diagonal entries are and its upper diagonal entries are . Then if and only if has either a row consisting entirely of zeros or all the row sums equal to zero.
Proof. Suppose that , where . Assume that . Then there exists such that and . Hence
So we must have (i) , (ii) , (iii) and (iv) either or . These boil down to having either or . Apply this argument to each row of to obtain the desired conclusion.