Let us consider the $2n \times 2n$ block matrix $\mathbf{X} = \pmatrix{\mathbf{P} & \mathbf{Q}\\ \mathbf{R} & \mathbf{S} }$ where $\mathbf{P} , \mathbf{Q} , \mathbf{R} , \mathbf{S}$ are square matrices of order $n$ which commute pairwise. Show that $\det \mathbf{X} = \det (\mathbf{PS} - \mathbf{QR}).$
The problem in this case I am having is when none of the matrices $ \mathbf{P} , \mathbf{Q} , \mathbf{R} , \mathbf{S} $ are invertible. Any help will be greatly appreciated.