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The manuscript presents the $LU$ approach to matrix biorthogonal polynomialswhen all the even ordered entries in the Gram matrix are zero. This arises incase of a quadratic transformation which is briefly discussed. Further, themain diagonal of the Gram matrix is a zero diagonal and we present the theorythat follows from this fact. Precisely, we discuss the Christoffeltransformation and matrix representations of the kernel polynomials, usuallycalled the ABC Theorem. Finally, we provide an illustration of our resultsassuming the Gram matrix has Hankel symmetry.

Quadratic transformation and matrix biorthogonal polynomials: an $\mathcal{LU}$ factorization approach