Let $d_A \colon \Omega^0(E) \longrightarrow \Omega^1(E)$ be a connection on a bundle $E\longrightarrow M$, and suppose its connection matrix is $\theta=(\theta_{ij})$ for some local frame of $E$. I know that the curvature matrix is given by $$ \Theta = d\theta + \theta \wedge \theta.$$
I want to prove that the curvature of the dual connection $d_{A^*}$ vanishes iff the curvature of $d_A$ vanishes.
But, if i take the dual coframe of the previous frame, I know that the connection matrix of $d_{A^*}$ is given by $-\theta^t$. Then the curvature matrix of the dual connection would be $$\Theta^* = - ( d\theta) ^t + (\theta\wedge\theta)^t$$ which needn't vanish. Am I missing something?