Work out all the details for the main properties of the determinant function [math]\det:M_N(\mathbb R)\to\mathbb R[/math], by using our geometric approach, and Thales.

Prove that for [math]H\in M_N(\pm1)[/math] we have [math]|\det H|\leq N^{N/2}[/math], with equality precisely when [math]H[/math] is Hadamard, in the sense that its rows are pairwise orthogonal.

Work out all the details for the properties of the determinant function [math]\det:M_N(\mathbb C)\to\mathbb C[/math], by using our algebraic approach, and permutations.

Learn the Jordan form, and come up with an alternative proof for our result stating that the diagonalizable matrices are dense.