Assume all algebras are finite dimensional quiver algebras over a field (no restriction of generaltiy if the field is algebraically closed).

Let A be a local Frobenius algebra. Is A isomorphic to its opposite algebra?

For non-Frobenius algebras this is false, see Do you know which is the minimal local ring that is not isomorphic to its opposite? (where the current question remained open, see the answer and comment).

Is there an easy example of a (not necessarily local) Frobenius algebra that is not isomorphic to its opposite algebra?