I have been struggling to derive inverse matrix of a $4 \times 4$ Lorenz matrix $\Lambda$. $ \Lambda = \begin{bmatrix} \gamma&0&0&-\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ -\beta \gamma & 0 & 0 & \gamma \end{bmatrix} $
My professor says that inverse is:
$ \Lambda^{-1} = \begin{bmatrix} \gamma&0&0&\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \beta \gamma & 0 & 0 & \gamma \end{bmatrix} $
Soo i tried to derive inverse matrix $\Lambda^{-1}$ using adjugate formula:
$\Lambda^{-1} = \frac{1}{|\Lambda|} \textrm{adj}(\Lambda)$
I did it quite metodically by first calculating the determinant of $\Lambda$, then matrix of minors, matrix of cofactors, adjugate matrix and in the end using the above formula to find the inverse (here is my whole derivation). Long story short, at the end I end up with this:
$ \Lambda^{-1} = \frac{1}{|\Lambda|} \textrm{adj}(\Lambda) = \frac{1}{\gamma^2 (1 - \beta^2)} \begin{bmatrix} \gamma & 0 & 0 &\beta \gamma\\ 0 & \gamma^2(1-\beta^2) & 0 & 0\\ 0 & 0 & \gamma^2(1-\beta^2) & 0\\ \beta \gamma & 0 & 0 & \gamma\\ \end{bmatrix} $
The result is not what my professor says i should get. In my adjugate matrix parts with $\gamma$ and $\beta \gamma$ seem wrong.
Is it possible my professor wrote down wrong inverse matrix? Could anyone point me to the right direction? I am kind of lost here, but i am sure i have done a lot of work and am near the solution.