multiplication table of matrices

Question

I would like to create a multiplication table of Pauli matrices to show that multiplying 2 Pauli matices results in a quaternion. What would be the best way to do that?

Answer 1

Thank you Rob Arthan

$\begin{array}{c|c|c|c} & \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} & \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} & \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \\\hline \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} & \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} & \begin{pmatrix} i&0\\ 0&-i \end{pmatrix} & \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} \\\hline \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} & \begin{pmatrix} -i&0\\ 0&i \end{pmatrix} & \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} & \begin{pmatrix} 0&i\\ i&0 \end{pmatrix} \\\hline \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} & \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix} & \begin{pmatrix} 0&-i\\ -i&0 \end{pmatrix} & \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \\ \end{array}$

$\begin{array}{c|c|c|c} & \sigma_1 & \sigma_2 & \sigma_3 \\\hline \sigma_1 & 1 & i & -j \\\hline \sigma_2 & -i & 1 & k \\\hline \sigma_3 & j & k & -1 \\ \end{array}$