# Show that $\{\sigma_i,\sigma_j\}=0, \text{if } i \neq j$

The following should be shown $$\{\sigma_i,\sigma_j\}=0, \text{if } i \neq j$$

This statements are given:

$$\{A,B\}=AB+BA$$ $$\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$

The question for me is how to show this in the best way. My first idea was to look at and multiply two general 2x2 matrixes and later sum up in reverse order (I mean this AB + BA, where A and B are general 2x2 matrices). But this approach did not really help me.

Another idea might be to simply test this expression for all three matrices (i.e. all combinations where $$i \neq j$$).

• There are only 3 pairs of $\{i,j\}$ to check; each check is two $2\times 2$ matrix multiplications. What stops you from doing it the direct way? – lisyarus Jun 20 '19 at 10:14