SIMON-MARAIS 2020 B1

Let $M$ be the set of $5 \times 5$ real matrices of rank 3. Given a matrix $A$ in $M$, 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 $M$.

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