Are the real and imaginary parts of an invertible matrix has to be invertible too?

Question

Let $\mathbf{A} \in \mathbb{C}^{N}$ be a complex matrix. Assuming that its inverse exists. Does it imply that both $\mathbf{A}_{R}^{-1}$ and $\mathbf{A}_{I}^{-1}$ exist? where $\mathbf{A} = \mathbf{A}_{R} + i\mathbf{A}_{I}$. If not, can you give a counter-example?

Answer 1

Certainly not: For any real matrix $\bf A$ (including invertible ones) viewed as a complex matrix, ${\bf A}_I = {\bf 0}$.