Strong commutativity preserving maps on subsets of matrices that are not closed under addition

Let Mn(D)Mn(D) be the ring of all n×nn×n matrices over a division ring DD, where n≥2n≥2 is an integer and let GLn(D)GLn(D) be the set of all invertible matrices in Mn(D)Mn(D). We describe maps f:GLn(D)→Mn(D)f:GLn(D)→Mn(D) such that [f(x),f(y)]=[x,y][f(x),f(y)]=[x,y] for all x,y∈GLn(D)x,y∈GLn(D). The analogous result for singular matrices is also obtained.