4x4 lorentz matrix and finding its inverse

1. Oct 7, 2012

71GA

I have been struggling to find an inverse to a Lorentz matrix $\Lambda$ using formula: $\Lambda^{-1}= \frac{1}{| \Lambda| }\textrm{adj}(\Lambda)$ from linear algebra.

$\Lambda = \begin{bmatrix} \gamma&0&0&-\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ -\beta \gamma & 0 & 0 & \gamma \end{bmatrix}$

My professor says that inverse to the matrix above is:

$\Lambda^{-1} = \begin{bmatrix} \gamma&0&0&\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \beta \gamma & 0 & 0 & \gamma \end{bmatrix}$

I tried to derive inverse matrix 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. And i end up with this:

$\Lambda^{-1} = \frac{1}{|\Lambda|} \textrm{adj}(\Lambda) = \frac{1}{\gamma^2 (1 - \beta^2)} \begin{bmatrix} \gamma & 0 & 0 &\beta \gamma\\ 0 & \gamma^2(1-\beta^2) & 0 & 0\\ 0 & 0 & \gamma^2(1-\beta^2) & 0\\ \beta \gamma & 0 & 0 & \gamma\\ \end{bmatrix}$

Well my 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?

I have tripple checked all the calculations, but i simply can't get the right result. Here is my whole derivation (please click the lower "DOWNLOAD" button).
Could anyone point me to the right direcction?
I am kind of lost here, but i am sure i have done a lot of work and am near the solution.

2. Oct 7, 2012

Muphrid

Erm, what do you think $\gamma^2 (1-\beta^2)$ is?

3. Oct 7, 2012

71GA

OMG it is 1!!! I AM SUCH A NOOB :)