Consider the following nonsingular block matrix: $$ M=\begin{pmatrix} A & B \\ -B^* & A^* \end{pmatrix} $$ where $A$ and $B$ are square matrices with the same dimension, $A^*$ and $B^*$ denote their complex conjugates.
Generating some random matrices for $A$ and $B$, I find that the determinant of $M$ is always a positive real number. Can anyone prove this fact, please?
From this property of a block matrix, and assuming that $A^*$ and $B^*$ commute:
$$\text{Det}(M) = \text{Det}(AA^*+BB^*)$$
Because both $AA^*$ and $BB^*$ are symmetric semi-positive, then $\text{Det}(M) \geq0$. Naturally, if you generate matrices at random, it is likely that they'll both be symmetric positive definite, and the determinant will be positive. But is you plug in two null matrices, the determinant will be zero. If $A$ and $B$ are non-null but have the same non-empty null space, then $M$ won't be zero, but still $\text{Det}(M)$ will be zero.