Determinant of a (square) matrix

The determinant of a matrix is a scalar value that factors into numerous properties of the matrix. It is also quite expensive to compute, involving operations for an matrix. (The determinant is only defined for square matrices.)

It has numerous definitions. Wikipedia says that the most common one is the “Leibniz definition,” but that involves a lot of math that I don’t know. ChatGPT tells me that we can take one of the formulas for calculating it, the Laplace expansion, as definitional. It is defined recursively:

Base case: matrix:

Inductive step ( matrix, ):

where is the submatrix obtained by removing the first row and the -th column of .