For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation.
Here $\det$ denotes the determinant.
Why can the antisymmetrizing bar be inserted in the middle because "there is already antisymmetry in the index lines"?
For an introduction to the notation, you can refer to Figures 12.17 and 12.18 below.