  The Matrix Bochner Problem aims to classify weight matrices $W$ such that its
algebra $\mathcal D(W)$, of all differential operators that have a sequence of
these matrix orthogonal polynomials as eigenfunctions, contains a second-order
differential operator. In [5] it is proven that, under certain assumptions, the
solutions to the Matrix Bochner Problem can be obtained through a
noncommutative bispectral Darboux transformation of some classical scalar
weights.
  The main aim of this paper is to introduce the concept of strong Darboux
transformation among weight matrices and explore the relationship between the
algebras $\mathcal{D}(W)$ and $\mathcal{D}(\widetilde{W})$ when $\widetilde{W}$
is a strong Darboux transformation of $W$. Starting from a direct sum of
classical scalar weights $\widetilde W$, and leveraging our complete knowledge
of the algebra of $\mathcal D(\widetilde W)$, we can easily determine the
algebra $\mathcal D(W)$ of a weight $W$ that is a strong Darboux transformation
of $\widetilde W$.
