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Let {Mathematical expression}, resp. {Mathematical expression} be the (given) invariant factors of the square matrices A, resp. B of order n over the ring of germs of holomorphic functions in 0 such that det A(λ)B(λ)≠0, λ≠0. A description of all possible invariant factors {Mathematical expression} of the product C=AB is given in the following cases: (i)β1 (or α1)≤2; (ii)β3 = 0 (α3= 0); (iii) α1-α2, β1βm≤1, α2+1-βm+1-0. These results, which hold for arbitrary n, are complemented with a few results leading to the description of all possible exponents γ1,γ2,γ3,γ4 for arbitrary α1,α2,α3,α4 β1,β2,β3,β4 in the case where the order n≤4

The local invariant factors of a product of holomorphic matrix functions: The order 4 case