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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.

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    ***Attention:*** This question had an accepted answer, but the answer contained *only* a picture, the link for which is now broken and the user who posted it has since been deleted. I have therefore deleted the answer so it is no longer marked as answered. KalEl, if you have the picture you got as an answer here, or can reconstruct the answer without the picture, please do post it in a new answer.2012-06-12
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    Please see the answer I have posted.2013-01-30

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