Special relativity: the transpose inverse of Lorentz matrix

Question

The exercise wants me to prove the relation: $$\Lambda_{\mu}^{\;\;\nu}\Lambda{^\mu}_{\;\alpha}=\delta^{\nu}_{\alpha}$$ And then conclude that $(\Lambda_{\mu}^{\;\;\nu})$ is the transpose inverse of $(\Lambda^{\mu}_{\;\;\nu})$. The first part I did well and can conclude that $$\Lambda_{\mu}^{\;\;\nu}=(\Lambda^{-1}){^\mu}_{\;\nu}\tag{1}$$ and, by definition $$(\Lambda^T)_{\mu}^{\;\;\nu}=\Lambda{^\nu}_{\;\mu}\tag{2}$$ I'm trying to, somehow, put $(2)$ in $(1)$ to get the answer, but I'm not sure if it's the right way or if assumed something wrong. Any help with it would be great.