New answers tagged matrices
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Snook, Counting bases of representable matroids shows that counting the number of maximal linearly independent subsets of columns is $\mathsf{\# P}$-complete. Your problem is probably also $\mathsf{\#P}$-complete. This suggests that you need an enumerative approach.
On the other hand, you can efficiently (in theory) approximate the number of maximal ...
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