Determinant/Multiplication theorem/Fact/Proof

Proof

We fix the matrix .

Suppose first that . Then, due to fact the matrix is not invertible and therefore, also is not invertible. Hence, .

Suppose now that is invertible. In this case, we consider the well-defined mapping

We want to show that this mapping equals the mapping , by showing that it fulfills all the properties which, according to fact, characterize the determinant. If denote the rows of , then is computed by applying the determinant to the rows , and then by multiplying with . Hence the multilinearity and the alternating property follows from exercise. If we start with , then and thus