I have been reading about TUM and my question is why the number of nonsingular $r \times r$ submatrices of the TU matrix $A$ of rank $r$ will give me the number of bases of $A$?
Recall that the definition of a TUM is as follows:
A rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix has determinant in $\{ 0, 1, -1 \}.$
Could anyone explain this to me please?
This statement is not correct in this form, unless I am misunderstanding the question. Consider the rank 1 totally unimodular matrix $$ A = \begin{pmatrix} 1&1 \\ 1 & 1 \end{pmatrix} $$ It has exactly 4 nonsingular submatrices and we can find two subsets of rows (or columns) that are bases of the row (or column) space of this matrix.