Prove that the matrix is totally unimodular

First, observe that since $R_1= - R_4$, $R_2= - R_5$, $R_3 = - R_6 $, then if any sub matrix uses any of these pairs of rows, then the determinant must be 0. Hence, (WLOG) we may assume that rows $1, 2, 3$ are removed (and replaced with rows 4, 5, 6 respectively in the sub matrix determinant calculation). Now call this matrix $A$.

If you look up the Wikipedia article, you will see a sufficient condition for Totally Unimodular matrices:

1. Every column of contains at most two non-zero entries.
2. Every entry in is 0, +1, or −1
3. If two non-zero entries in a column of $A$ have the same sign, then the row of one is in $B$, and the other in $C$.
4. If two non-zero entries in a column of $A$ have opposite signs, then the rows of both are in $B$, or both in $C$.

Observe that if $B$ is the set of rows 4, 5, 6 and $C$ is the set of rows 7, 8, 9, 10, then this will satisfy the conditions. Hence we are done.