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 associated to a matrix , in other words
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 ; 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.