Prove that if $M \in M_{n,n}(F)$ can be written in the form $M=\begin{pmatrix} A & B\\ O & C \end{pmatrix}$, where $A$ and $C$ are square matrices, then $\det(M)=\det(A)*\det(C)$
I am not sure the rigorous way of proving, and I see some proof goes by block matrix, which is something not yet learned about.
My idea is constructing a triangular matrix, then by cofactor expansion, we lead to a linear combination of the form of the determinant. Is that doable?
Another thing is should I separate by cases, when the matrix is invertible or when not invertible?