I was reviewing the Wikipedia page for determinants which states that for all complex matrices, the determinant is the product of its eigenvalues. However, isn't this true for real matrices as well, if we just consider them in the larger vector space $\mathbb{C^{n\times n}}$? Looking at the complex Jordan form (which won't necessarily exist in $\mathbb{R}$) it seems to be true.
Is there anything I am missing that would prevent the determinant of a matrix $A \in \mathbb{R^{n\times n}}$from being the product of its real and complex eigenvalues, like could there be an odd number of complex eigenvalues for a real matrix?