Let 𝒞 be a uniform clutter and let A be the incidence matrix of 𝒞. We denote the column vectors of A by v
1,…,v
q
. Under certain conditions we prove that
𝒞 is vertex critical. If 𝒞 satisfies the max‐flow min‐cut property, we prove that A diagonalizes over ℤ to an identity matrix and that v
1,…,v
q
form a Hilbert basis. We also prove that if 𝒞 has a perfect matching such that 𝒞 has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ℤ to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v
1,…,v
q
is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion‐freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.
