Is the matrix totally unimodular? Let $A=\begin{bmatrix}1&0&1&1&0&0&1&0&0\\0&1&0&1&0&0&0&0&0& \\0&0&1&1&1&0&0&0&0\\0&0&-1&-1&0&1&-1&0&0\\0&0&0&-1&0&0&0&1&0\\0&0&-1&-1&0&0&0&0&1\end{bmatrix}$, is the matrix totally unimodular?
I realised that if I change the order of columns in A I could get  change the matrix I would get  $A=( B I_{6x6})$, but I wasn't sure if I could use this for anything.

Successively applying column-swapping, $C_3 \leftrightarrow C_5$, $C_4 \leftrightarrow C_6$, $C_7 \leftrightarrow C_8$ and $C_8 \leftrightarrow C_9$, we get,
$A^{\prime}=\begin{bmatrix}1&0&0&0&0&0&1&1&1\\0&1&0&0&0&0&0&1&0& \\0&0&1&0&0&0&1&1&0\\0&0&0&1&0&0&-1&-1&-1\\0&0&0&0&1&0&0&-1&0\\0&0&0&0&0&1&-1&-1&0\end{bmatrix}=\left[I_6 \mid B\right]$, where $B=\begin{bmatrix}1&1&1\\0&1&0& \\1&1&0\\-1&-1&-1\\0&-1&0\\-1&-1&0\end{bmatrix}$
Now, from here, we have

*

*$A$ is unimodular iff $A^{\prime}$ is unimodular.

*$A^{\prime}$ is unimodular iff $B$ is unimodular.

Now, all remains to be shown is that $B$ is unimodular.

*

*All elements of $B \in \{-1,0,1\}$

*Any $2 \times 2$ submatrix of $B$ is of the form $\begin{bmatrix}x&x\\ x&x\end{bmatrix}$  or $\begin{bmatrix}x&x\\ -x&-x\end{bmatrix}$ or $\begin{bmatrix}y&0\\ x&x\end{bmatrix}$ or $\begin{bmatrix}0&y\\ x&x\end{bmatrix}$ or $\begin{bmatrix}x&x\\ y&0\end{bmatrix}$ or $\begin{bmatrix}x&x\\ 0&y\end{bmatrix}$ or $\begin{bmatrix}x&y\\ x&0\end{bmatrix}$ or $\begin{bmatrix}y&x\\ 0&x\end{bmatrix}$, or their transpose matrices, where $x,y \in \{-1,0,1\}$, clearly $det(B) = 0$ or $xy \in \{-1,0,1\}$

*Finally, show that all $3 \times 3$ submatrices (there are only ${6 \choose 3} = 20$ of them, each one having all cofactors $\in \{-1,0,1\}$) have determinant $\in \{-1,0,1\}$.

