The two most elementary ways to prove an N x N matrix's determinant = 0 are: A) Find a row or column that equals the 0 vector. B) Find a linear combination of rows or columns that equals the 0 vector. A can be generalized to C) Find a j x k submatrix, with j + k > N, all of whose entries are 0. My minor question is: Is C a named theorem that one can easily reference? My major question is: Are there are other canonical ways of proving a determinant = 0? The context is that I'm trying to solve the generalized form of what was, as stated, a very easy Putnam Exam problem, and I last took a linear algebra class in 1974. *In response to comments below, let me say: - Thanks! - This isn't about computational efficiency. - Frobenius-Koenig looks very helpful.*