Does cofactor expansion generalize to complex matrices?

Question

When finding the determinant of some $n * n$ matrix $A$ when $$\forall i,j\in\mathbb{N} ,i\leq n\land j\leq n\implies A_{ij} \in \mathbb{C}$$ Can cofactor expansion be used under the normal definition of complex multiplication?

Answer 1

On the face of it, your question is "is this a correct way to calculate the determinant of a complex matrix", but to make sense of that question you first have to define what the determinant of a complex matrix is. You could define the determinant by co-factor expansions. What you really mean is, does the object thus defined resemble the determinant of a real matrix.

The answer is yes. This is why a lot of authors prefer to do linear algebra in the context of a vector space over a field, rather than specifically over the real numbers. As long as you can describe a field containing the entries of your matrix, then we can define a function on matrices using co-factor expansions, and it will satisfy all the usual properties of the determinant.