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I conjecture the following.

Given three rectangular matrices $A$, $B$ and $C$ such that the following two block matrices $ \begin{bmatrix} A & B \\ \end{bmatrix} $ and $ \begin{bmatrix} A \\ C \\ \end{bmatrix} $ are totally unimodular, the following matrix $\begin{bmatrix} A & B \\ C & 0 \\ \end{bmatrix} $ is totally unimodular.

Can you help me to prove it?

1 Answers 1

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Here is a counterexample, i'm sorry.

$ \det \left[ \begin{array}{c c | c} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \hline -1 & -1 & 0 \end{array} \right] = 2 $

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    Indeed, perhaps we can use Laplace expansion, in this case. Let me think a little bit more. – 2017-02-12
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    @maxdan94 0-matrix can also be rectangular in your case? – 2017-02-12
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    @maxdan94 I changed a bit my answer in order to cover the case of not square B' – 2017-02-13
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    no, currently, it doesn't work. – 2017-02-13
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    This discussion is about my old answer, should we consider to remove the comments ? – 2017-02-13