In this article we present several matrix theorems. The first of these, the theorem on Mondriga matrices, originated in the study of the immobilization of plane and solid figures in Euclidean space by points on their boundary, as posed by Kuperberg and Papadimitriou. In this context the theorem provides the tool to prove that if a Letrahedron satisfies a first-order immobilization condition, it also satisfies a second-order immobilization condition [2]. The first proofs obtained involved the study of many cases (given by the faces of n-dimensional polyhedra). Then we found a characteristic polynomial factorization which gave the result simply. This factorization in turn yielded several seemingly unrelated theorems.