Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.