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4x4 lorentz matrix and finding its inverse

  1. Oct 7, 2012 #1
    I have been struggling to find an inverse to a Lorentz matrix [itex]\Lambda[/itex] using formula: [itex]\Lambda^{-1}= \frac{1}{| \Lambda| }\textrm{adj}(\Lambda)[/itex] from linear algebra.

    [itex]
    \Lambda =
    \begin{bmatrix}
    \gamma&0&0&-\beta \gamma \\
    0 & 1 & 0 & 0\\
    0 & 0 & 1 & 0\\
    -\beta \gamma & 0 & 0 & \gamma
    \end{bmatrix}
    [/itex]

    My professor says that inverse to the matrix above is:

    [itex]
    \Lambda^{-1} =
    \begin{bmatrix}
    \gamma&0&0&\beta \gamma \\
    0 & 1 & 0 & 0\\
    0 & 0 & 1 & 0\\
    \beta \gamma & 0 & 0 & \gamma
    \end{bmatrix}
    [/itex]

    I tried to derive inverse matrix metodically by first calculating the determinant of [itex]\Lambda[/itex], 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:

    [itex]
    \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}
    [/itex]

    Well my result is not what my professor says i should get. In my adjugate matrix parts with [itex]\gamma[/itex] and [itex]\beta \gamma[/itex] 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. jcsd
  3. Oct 7, 2012 #2
    Erm, what do you think [itex]\gamma^2 (1-\beta^2)[/itex] is?
     
  4. Oct 7, 2012 #3
    OMG it is 1!!! I AM SUCH A NOOB :)
     
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