Determinant/Recursively/Multilinear/Fact/Proof

Proof

Let

where we denote the entries and the matrices arising from deleting a row in an analogous way. In particular, and . We prove the statement by induction over ,  For , we have and

due to the induction hypothesis. For , we have and . Altogether, we get

The compatibility with the scalar multiplication is proved in a similar way, see exercise.