On commuting integer matrices

Given $d,N\in \mathbb{N}$, we define ${\mathfrak{C}}_d(N)$ to be the number of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N]\cap \mathbb{Z}$ such that $AB=BA$. We prove that $$N^{10}\ll {\mathfrak{C}}_3(N)\ll N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further establish that $${\mathfrak{C}}_2(N)=K(2N+1{)}^5(1+o(1)),$$ where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity.