# 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 $$i \neq j$$).

So I'm curious about your opinions, how would you show that?

PW: The task comes from a book on quantum computer science that I am currently reading.

Additional Keywords: anticommutator, commutator

• 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
• @lisyarus I fully agree – Peter Jun 20 '19 at 10:14
• OK, I almost thought that would be the end of it. Then I thank you for the assessment! Thanks. I have given you a +1, because I agree with your opinions :) – P_Gate Jun 20 '19 at 10:15