Universal equivalence of general and special linear groups over fields

In this paper, we study universal equivalence of general and special linear groups over fields. We give the following criterion for this relation to hold: two groups $\mathbf G_n(K)$ and $\mathbf G_m(L)$ ($\mathbf G=\mathrm{GL}, \mathrm{SL}$, $K$ and $L$ are infinite fields) are universally equivalent if and only if $n=m$ and the fields $K$ and $L$ are universally equivalent.