Let script F sign be a Banach space with a normalized, 1-unconditional basis. Each operator on the script F sign-direct sum of a sequence (x_i)_i∈ℕ of Banach spaces corresponds to an infinite matrix. We study whether this correspondence is multiplicative, in which case we say that matrix multiplication works. We prove that matrix multiplication works if at least one of the following two conditions is satisfied: (i) for each i ∈ ℕ, each operator from x_i to script F sign is compact; (ii) the basis of script F sign is shrinking and, for each i ∈ ℕ, each operator from script F sign to x_i is compact. In the case where script F sign is either c₀ or ℓ_p, where 1 ≤ p < ∞, the converse also holds.