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Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating v and w , then there exists f ∈ S separating v and w. In this article we consider the action of G = SL 2 (C) × SL 2 (C) on the C -vector space M 2 , 2 n of n -tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S n of C [ M 2 , 2 n ] G are known, and | S n | = 1 24 (n 4 − 6 n 3 + 23 n 2 + 6 n). In recent work, Domokos [8] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 , 2 n ] G. Our main result shows that any separating set for C [ M 2 , 2 n ] G has cardinality ≥ 5 n − 9. In particular, there is no separating set of size dim ⁡ (C [ M 2 n ] G) = 4 n − 6 for n ≥ 4. Further, S 4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S n. We also consider the action of G = SL l (C) on M l , n by left multiplication. In that case the algebra of invariants has a minimum generating set of size ( n l ) (the l × l minors of a generic matrix) and dimension l n − l 2 + 1. We show that a separating set for C [ M l , n ] G must have size at least (2 l − 2) n − 2 (l 2 − l). In particular, C [ M l , n ] G does not contain a separating set of size dim ⁡ (C [ M l , n ] G) for l ≥ 3 and n ≥ l + 2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem. [ABSTRACT FROM AUTHOR]