**Question.** Let $K$ be a field (assume $K=\mathbb{C}$ if this simplifies the problem). What is the right global dimension of the $K[x,y]$-algebra:
$$
A=\left[\begin{array}{cc}
K[x,y] & xK[x,y] \\
K[x,y] & K[x,y]
\end{array}\right] .
$$
I am not an expert in homological algebra so I would also appreciate advice on how to analyze such examples in general.

**Some Motiativation.** A $1$-variable counterpart of this example, i.e., the $K[x]$-algebra
$$
B=\left[\begin{array}{cc}
K[x] & xK[x] \\
K[x] & K[x]
\end{array}\right] .
$$
has right global dimension $1$, or equivalently, it is *hereditary*. In fact, loosely speaking, hereditary $K[x]$-orders in $\mathrm{M}_{n\times n}(K(x))$ all look roughly like this example. By analogy, I would expect the ring $A$ to have right global dimension $2$.

**Remark.** The rings $A$ and $B$ are noetherian, so their right and left global dimensions coincide.