  The manuscript presents the $LU$ approach to matrix biorthogonal polynomials
when all the even ordered entries in the Gram matrix are zero. This arises in
case of a quadratic transformation which is briefly discussed. Further, the
main diagonal of the Gram matrix is a zero diagonal and we present the theory
that follows from this fact. Precisely, we discuss the Christoffel
transformation and matrix representations of the kernel polynomials, usually
called the ABC Theorem. Finally, we provide an illustration of our results
assuming the Gram matrix has Hankel symmetry.
