Exercise from Supergravity textbook

[Korybut]

Homework Statement

In Ex. 2.4 from textbook Supergravity by Freedman and Van Proeyen one needs to prove the following identity
$$ A^\dagger \sigma_\mu A=\sigma_\nu \Lambda^\nu{}_\mu $$

Homework Equations

It is easy to prove the other identity in this exercise
$$ A\bar{\sigma}_\mu A^\dagger=\bar{\sigma}_\nu \Lambda^{-1} {}^\mu{}_\nu $$

The Attempt at a Solution

I proved the first identity but when sigma matrices are substituted by barred ones which means inversion of the zero component (metric in this book is mostly plus)

Is it a typo or I can't see something pretty easy?

 

[mark726]

I would first double check the equations and make sure there are no typos or mistakes in the book. If there are no errors, then I would try to approach the problem from a different angle and see if there is a different way to prove the identity. I would also consult with my colleagues or do some research to see if there are any known solutions or techniques for proving this type of identity. It is possible that there is a simple solution that I am missing, so I would also try to think outside of the box and look at the problem from different perspectives. Overall, I would remain persistent and determined to find a solution to the problem.