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Let $I$ be the ideal in $A\otimes A$ generated by elements of the form $a\otimes 1-1\otimes a$. (Equivalently, this is the kernel of the multiplication map $A\otimes A\to A$.) We will assume that $0\neq A\neq k$, so that $0<I<A\otimes A$. As everything is finite-dimensional, the sequence of powers $I^n$ must eventually stabilise. Let $n$ be least such that $I^n=I^{n+1}=I.I^n$. By the standard determinant trick, this gives an element $u$ with $u=1\pmod{I}$ and $u.I^n=0$. Let $m$ be least such that $u.I^m=0$. As $u=1\pmod{I}$ we have $u\neq 0$, so $m>0$. Now $u.I^{m-1}\neq 0$ and $u.I^{m-1}\leq\text{ann}(I)$ so $\text{ann}(I)\neq 0$.