The Determinant

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The determinant summarizes how much a linear transformation, from a vector space to itself, “stretches” its input.

The determinant of a matrix exists whenever the matrix is square, and whenever entries can be added and multiplied, assuming the multiplication is commutative. Thus determinants exist for more than just matrices of numbers; for example, a square matrix whose entries are real-valued functions has a well-defined determinant.

There are different ways to define it. We present two, and record some of its basic properties.