Rank of a Complex matrix in terms of its real and imaginary parts

Question

Let $A\in\mathbb{C}^{n\times n}$ be a complex matrix (i.e., with complex entries). We separate the real and imaginary parts of $A$ as follows: $A=A_1+iA_2$ where $A_1,A_2\in\mathbb{R}^{n\times n}$. Is there any relation between the rank or determinant of $A$, $A_1$, and $A_2$? What should be the condition on $A_1,A_2$ such that $A$ will be non-singular?

Thanks.