  The paper is devoted to 2-local derivations on matrix algebras over unital
semi-prime Banach algebras. For a unital semi-prime Banach algebra
$\mathcal{A}$ with the inner derivation property we prove that any 2-local
derivation on the algebra $M_{2^n}(\mathcal{A}),$ $n\geq 2,$ is a derivation.
We apply this result to $AW^\ast$-algebras and show that any 2-local derivation
on an arbitrary $AW^\ast$-algebra is a derivation.
