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So I can use the following to get an equation for the inverse:

[tex]x^{\overline{\mu}}x_{\overline{\mu}}=\Lambda^{\overline{\mu}}_{\;\alpha}x^{\alpha}\Lambda^{\beta}_{\;\overline{\mu}}x_{\beta}[/tex]

And therefore

[tex]\Lambda^{\beta}_{\;\overline{\mu}}\Lambda^{\overline{\mu}}_{\;\alpha}=\delta^{\beta}_{\;\alpha}[/tex]

This equation is just the one in ch2 from Schutz. But I can just as well reason as follows:

[tex]x^{\overline{\mu}}x_{\overline{\mu}}=\eta_{\overline{\mu}\overline{\nu}}x^{\overline{\mu}}x^{\overline{\nu}}=\eta_{\overline{\mu}\overline{\nu}}\Lambda^{\overline{\mu}}_{\;\alpha}x^{\alpha}\Lambda^{\overline{\nu}}_{\;\beta}x^{\beta}=\Lambda^{\overline{\mu}}_{\;\alpha}x^{\alpha}\Lambda_{\overline{\mu}\beta}x^{\beta}[/tex]

And therefore

[tex]\Lambda_{\overline{\mu}\beta}\Lambda^{\overline{\mu}}_{\;\alpha}=\eta_{\beta\alpha}[/tex]

Or

[tex]\Lambda_{\overline{\mu}}^{\;\ \beta}\Lambda^{\overline{\mu}}_{\;\alpha}=\delta^{\beta}_{\;\alpha}[/tex]

Taken together, we seem to have

[tex]\Lambda_{\overline{\mu}}^{\;\ \beta}=\Lambda^{\beta}_{\;\overline{\mu}}[/tex]

Is this correct? It seems wrong to me, and it seems that I might’ve confused my tensor and matrix indices, I’m just not sure how...