Exercise 1, § 9

Let $K$ be a commutative field with at least 3 elements and $A$ the algebra over $K$ with a basis consisting of the unit element 1 and two elements $e_1, e_2$ such that $e_1^2 = e_1, e_1 e_2 = e_2, e_2 e_1 = e_2^2 = 0$; let $A^0$ be the opposite algebra to $A$. Show that there exists in $A$ elements $x$ such that $\operatorname{Tr}_{A/K}(x) \neq \operatorname{Tr}_{A^0/K}(x)$ and $N_{A/K}(x) \neq N_{A^0/K}(x)$.