Fermion quantization derivation in quantum field theory

[Andrew3]

Homework Statement

I am studying the part of Fermion Quantization in QFT. However, I am puzzled about the derivation in one class in youtube.
https://www.youtube.com/watch?v=FNxespJgDDE&list=PLbMVogVj5nJQ3slQodXQ5cSEtcp4HbNFc&t=826s
At 13:46, why does (sigma^3)_a^c (sigma^1)_{cb}=(sigma^3 sigma^1)_{ab}?

Relevant Equations

sigma^0 is the unit. sigma^k is the pauli matrices (k=1,2,3). 'a','b' and 'c' denotes the indexes of matrices elements.

 

[Antarres]

Because contraction of the $c$ index is just an index form of noting matrix multiplication. What he wrote is just matrix multiplication law for those matrices. For example, the element of product of two matrices is:
$$(AB)_{mn} = \sum_{k}A_{mk}B_{kn}$$
In the notation on the board, the sum is implied over indices that appear twice(that's Einstein summation convention), and also lifting and lowering indices is probably trivial since metric is trivial.