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We prove that any product of two non-abelian free groups, $\Gamma=\mathbbF_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. Thismeans that there exist asymptotic representations $\pi_n:\Gamma\rightarrow\text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which arenot close to actual representations. As a consequence, we prove the existenceof contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$,with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close toany matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. This settlesin the negative a natural version of a question concerning almost commutingmatrices posed by Rosenthal in 1969.

Almost commuting matrices and stability for product groups