Determinant/Transposed matrix/Universal property/Fact/Proof

Proof

If is not invertible, then, due to fact, the determinant is and the rank is smaller than . This does also hold for the transposed matrix, so that its determinant is again . So suppose that is invertible. We reduce the statement in this case to the corresponding statement for the elementary matrices, which can be verified directly, see exercise. Because of fact, there exist elementary matrices such that

is a diagonal matrix. Due to exercise, we have

and

The diagonal matrix is not changed under transposing it. Since the determinants of the elementary matrices are also not changed under transposition, we get, using fact,