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We investigate the statistical properties of $C=uvu^{-1}v^{-1}$, when $u$ and$v$ are independent random matrices, uniformly distributed with respect to theHaar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived forthe average value of power sum symmetric functions of $C$, and also forproducts of the matrix elements of $C$, similar to Weingarten functions. Thedensity of eigenvalues of $C$ is shown to become constant in the large-$N$limit, and the first $N^{-1}$ correction is found.

Commutators of random matrices from the unitary and orthogonal groups