Commutativity up to scalar implies commutativity in an algebra

Let $A$ be a (not necessarily commutative) algebra over a field $k$. Suppose that for all $a,b\in A$, we have $kab=kba$, i.e. commutativity up to scalar. Show that then $A$ is commutative.

In the assumption, it is important that it holds for all $a,b\in A$, otherwise it would be false. This is a step in Exercise 2.4.8 of Radford's book "Hopf algebras".

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