What is a determinant?

Inspired by this Reddit thread which was inspired by this MathOverflow thread. Prerequisites: matrix multiplication. The original version of the post was missing the minus sign in the definition of the determinant; thanks to my friend Josh Zelinsky for politely pointing that out.

A determinant is the real number \det A associated to a n \times n matrix A = (a_{i,j}), in other words

A = \left[ \begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{array} \right],

such that

\det A = \displaystyle \sum_{\sigma \in S^n} (-1)^{\text{sgn}(\sigma)} a_{1,\sigma(1)} a_{2,\sigma(2)} \cdots a_{n, \sigma(n)}.


This definition, while correct and more concise than the expansion-by-cofactors algorithm used to compute it (which I gave as the definition in MTH 237 because like fun am I introducing permutations to use once in a month-long course), is completely nonilluminating. In English, the definition says that you multiply together every entry in a path across the matrix such that in each path you never repeat rows or columns and add the results. (One such product would be a_{1,2}a_{3,4}a_{2,3}a_{4,1}; notice each row and column is used only once.) This still tells us nothing about what a determinant means or why one gives a shit, and since determinants are one of the most useful tools by which to understand linear operators / square matrices, it’s a pretty big gap to ignore.

It is necessary to add an interpretation to our (definition, algorithm) pair.

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