MathDB

Let

\mathcal{M}

be the set of

5\times 5

real matrices of rank

3

. Given a matrix in

\mathcal{M}

, the set of columns of

A

has

2^5-1=31

nonempty subsets. Let

k_A

be the number of these subsets that are linearly independent.
Determine the maximum and minimum values of

k_A

, as

A

varies over

\mathcal{M}

. The rank of a matrix is the dimension of the span of its columns.

Problem Statement