# Special relativity: the transpose inverse of Lorentz matrix

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.