I am told to show that these matrices are anticommutative (AB = -BA)$$
\beta=\begin{bmatrix}I\_2&0\\0&-I\_2\end{bmatrix}, \alpha\_x=\begin{bmatrix}0&\sigma\_1\\\sigma\_1&0\end{bmatrix},\alpha\_y=\begin{bmatrix}0&\sigma\_2\\\sigma\_2&0\end{bmatrix}, \alpha\_z=\begin{bmatrix}0&\sigma\_3\\\sigma\_3&0\end{bmatrix}$$ where $\sigma\_1=\begin{bmatrix}0&1\\1&0\end{bmatrix}, \sigma\_2=\begin{bmatrix}0&-i\\i&0\end{bmatrix}, \sigma\_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ (the pauli spin matrices). I could just use brute force and plug every matrix in for B and A, but my professor is very hard on us when it comes to proofs and only gives full credit for "elegant" proofs. The only problem is, I have no idea how I would even start to prove this; any help or hints would be greatly appreciated!!!