The determinant is a function that takes a square matrix as an input and produces a scalar as an output. So unlike a vector space, it is not an algebraic structure. However, it has many beneficial properties for studying vector spaces, matrices and systems of equations, so it is hard to ignore (though some have tried). While the properties of a determinant can be very useful, they are also complicated to prove.

Section DM Determinant of a Matrix

Subsection EM: Elementary Matrices

Subsection DD: Definition of the Determinant

Subsection CD: Computing Determinants

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section PDM Properties of Determinants of Matrices

Subsection DRO: Determinants and Row Operations

Subsection DROEM: Determinants, Row Operations, Elementary Matrices

Subsection DNMMM: Determinants, Nonsingular Matrices, Matrix Multiplication

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Subsection EM: Elementary Matrices

Subsection DD: Definition of the Determinant

Subsection CD: Computing Determinants

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section PDM Properties of Determinants of Matrices

Subsection DRO: Determinants and Row Operations

Subsection DROEM: Determinants, Row Operations, Elementary Matrices

Subsection DNMMM: Determinants, Nonsingular Matrices, Matrix Multiplication

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions