If $A, B \in \{0, 1, −1\}m×n$ are totally unimodular $(TU)$ matrices, then show that the matrix
\begin{bmatrix} A&0\\ 0&B\end{bmatrix}
is also $TU$, where $O \in \{0\}$ m×n is an all $0$’s matrix.
I am trying to solve the above problem, but I'm not sure how one proves a matrix is $TU$. I've seen a few proofs online that involve induction with certain property assumptions about the matrix but I'm not sure how to apply this here. It is easy to see that [$A$ $0$] and [$0$ $B$] are $TU$ because you will always have a column of $0$s and that gives you a determinant of $0$ or you have a submatrix of $A$ or $B$ and that itself is $TU$ but I'm not sure how to proceed from there. Is there a general way I can show a matrix is unimodular, and how can i show this final matrix is $TU$?