The question asks if this matrix is totally unimodular or not(put into table form to look a bit clearer):
1 0 0 0 0 0 0 1 -1 0 1 0 -1 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 1 -1 0 1 0 -1 -1 0 0 0 1 0 0 0 1 0 0 0 0 0 0
So the definition of a matrix being totally unimodular is that all square submatrices have determinant 1,-1 or 0. This would mean calculating the determinant of all 1x1 (straightforward) matrices, 2x2, 3x3, 4x4, 5x5 and 6x6 (slightly less straighforward and significantly more time-consuming!)
Is there any theorems or instant shortcuts I can use to show whether this matrix is totally unimodular or not?