2-Local derivations on matrix algebras and algebras of measurable operators

Let $\\mathcal{A}$ be a unital Banach algebra such that any Jordan derivation from $\\mathcal{A}$ into any $\\mathcal{A}$-bimodule $\\mathcal{M}$ is a derivation. We prove that any 2-local derivation from the algebra $M_n(\\mathcal{A})$ into $M_n(\\mathcal{M})$ $(n \\geq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.