I must be doing something wrong ?

Main Question or Discussion Point

I've come here directly to the physics subforum for help understanding something.

Now I am really confused: consider the matrix form of $$a^k$$ and calculate it all out we have

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix}$$

A nullified matrix?

Have I got my $$a^k$$ matrix right... ?

I just don't understand why the relationship

$$\beta \alpha^k = \gamma^k$$

would be important if it spat out a zero matrix, which makes me wonder strongly whether I even have the right conditions down for $$a^k$$?

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Sorry, terrible mistake on my part, I recalculated it properly this time

$$\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & -1 & 0 & 0 \\-1 & 0 & 0 & 0 \end{pmatrix}$$