Variable n' : nat.
Local Notation n := n'.+1.
Canonical matrix_algType :=
Eval hnf in AlgType R 'M[R]_n (fun k => scalemxAr k).
End MatrixAlgType.

(* The theory of determinants *)
Lemma determinant_multilinear n (A B C : 'M[R]_n) i0 b c :
row i0 A = b *: row i0 B + c *: row i0 C ->
row' i0 B = row' i0 A ->
row' i0 C = row' i0 A ->
\det A = b * \det B + c * \det C.