Show that this matrix is Totally unimodular

Question

Suppose we know that $A \in R^{n\times n}, B\in R^{n\times n}, \begin{bmatrix}a \\ A\end{bmatrix}\in R^{(n+1)\times n}$ and $[b \space B]\in R^{n\times(n+1)}$ are all total unimodular matrix. How can we show that $\begin{bmatrix}1 & a^Tb \\ ab^T & AB\end{bmatrix}$ is also TUM? Thanks!