Is is possible to find the absolute value of the determinant of a complex matrix M, given two real matrices A & B of the same dimension of M; where for N = 2,
M would be: $\begin{matrix} a_{00} + i*b_{00} & a_{01} + i*b_{01} \\ a_{10} + i*b_{10} & a_{11} + i*b_{11} \end{matrix}$?
I.e. using just the real numbers and determinants of A & B is there any known relation that can get me the absolute value of the complex determinant from M? Thanks!
$A$ and $B$ can be any matrix, really. Your question is somehow equivalent to; is there a shorter formula for $\det (A+B)$ where $A, B \in \mathcal{M_n}(\mathbb{R})$ ?
Since the determinant application is only a group homomorphism from the general linear group of degree $n$ to $(\mathbb{R}^*, \times)$, meaning $\det(AB) = (\det A)( \det B)$ and $\det(A^{-1}) = (\det A)^{-1}$, there is no general formula for the determinant of the sum of two matrices.