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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.

2 added 148 characters in body

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.

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# Proving a determinant = 0

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.