In order to study Brauer’s orthogonal centralizer algebra, the authors consider certain quadratic matrices of very big size. These matrices correspond to equivariant endomorphisms of the representation space of the product of two symmetric groups. Hence, using representation- theoretic methods, they can be transformed into a simpler form. This fact is used to compute the determinant and the rank of the given matrices. After presenting the results of the computations, the authors state a number of conjectures.