# Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty :

Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square matrix $M = \begin{bmatrix} A & X\\ X^\top & B \end{bmatrix}$ such that $\det M$ is related to the number of non zero columns of $X$ ?

All my attempts failed, but perhaps I was looking in a wrong direction...

-
How should that work? Can you give an example? –  draks ... Sep 27 '12 at 9:06
The matrices $A$ and $B$ could be anything, for example an empty matrix $A$ and identity for $B$. And by a "relation" between $det M$ and $nzc(X)$ the number of non zero columns of $X$, I meant $det M = f(nzc(X))$ with any monotonic function $f$. –  Emile Sep 27 '12 at 9:11