Determinant of complex matrix

Question

How is the determinant of a complex matrix calculated?
Is it the same algorithm as for real matrices, but the determinant itself is complex instead of real?

(I was unable to find any hints with google...)

Answer 1

The determinant is calculated in the same way, regardless of what field the numbers in the elements of the matrix are taken from.

Answer 2

The type signature of the determinant is indeed not prominently displayed in the english language Wikipedia article on determinants, hit #3 in my google search, after the German Duden wordbook entry and the German language article.

It says

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

in the introduction which hints on the result type, if you consider algebraic closure.

On the other hand the article on multilinear forms does: $$ f: V^n \to K $$ for $V$ being a $n$-dimensional vector space over $K$.

Of course you need to be familiar with the Weierstraß characterisation of the determinant as a (normed) alternating multilinear form to look there, which is not linked in the definition in the article on determinants and is buried more deep down in that text.

Kind of surprising for a basic article which should have gotten many eyeballs already.