Suppose that $A$ is a finite dimensional $k$-algebra. Call $Q=\mathrm{Hom}_k(A,k)$. $Q$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial extension of $A$ is defined as follows:
the underlying vectorspace is $T(A)=A\oplus Q$ and the multiplication is given by
$(a,q)(a^\prime,q^\prime)=(aa^\prime,aq^\prime+qa^\prime)$ for $a,a^\prime\in A$ and $q,q^\prime\in Q$.
I want to prove that $T(A)$ is Frobenius (see here for a definition). If we assume to know that the answer to the question linked above is true then we have to prove that $T(A)$ is selfinjective. Any idea how can I prove it?