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. More later as I work through them- Frobenius-Koenig looks very helpful.