I have an $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $$M = ( x I - A z -B w \mid y I - C z - D w).$$ Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let $I$ be the ideal of maximal minors of $M$. I believe that the CM-regularity of this ideal is $m$. However I have no idea how to show it.

This is not a generic determinantal ideal, so many results for them do not apply in this case. However maybe there is a way to specialize and still keep the CM-regularity.

In general I want to show that for matrices of different size the closed subschemes of $\mathbb{P}^3$ defined by the ideals are non-isomorphic. Under certain assumptions the subscheme will be one dimensional.

I'm not very familiar with results about the CM-regularity (outside Eisenbud's "Commutative Algebra"), so a reference of something will be appreciated as well.

Thanks.