# determinant

(4.8 hours to learn)

## Summary

The determinant is a scalar value associated with a square matrix. It is convenient algebraically because it behaves nicely with respect to matrix multiplication, inverses, and transposes, as well as the Gaussian elimination operations. It gives the factor by which volumes are rescaled by the matrix's associated linear transformation. It also equals the product of the eigenvalues.

## Context

This concept has the prerequisites:

- vectors
- matrix transpose (the matrix transpose preserves the determinant)
- Gaussian elimination (The Gaussian elimination operations do nice things to the determinant.)
- matrix multiplication (The determinant of a product is the product of the determinants.)
- matrix inverse (The determinant of the inverse is the inverse of the determinant.)

## Goals

- Define the determinant (in terms of a sum over column permutations)

- Be able to calculate the determinant recursively in terms of cofactors

- Know what the Gaussian elimination operations do to the determinant of a matrix

- Be able to manipulate the determinant algebraically, together with matrix multiplication, inverse, and transpose

- Derive the identities relating the determinant to matrix multiplication, inverse, and transpose

- Show that a matrix is invertible if and only if its determinant is nonzero

- Show that the determinant of a triangular matrix is the product of the diagonal entries

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Sections 1.5, "Introduction to determinants and the cross product," pages 43-50, and 7.5, "Determinants and n-dimensional volume," pages 309-321

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Sections 5.1, "The properties of determinants," and 5.2, "Permutations and cofactors," pages 244-263

## See also

- The determinant is used to define:
- the characteristic polynomial of a matrix, which is important for dealing with eigenvalues
- multivariate Gaussians , a widely used probability distribution for multivariate data

- the constant term of the characteristic polynomial
- the product of the eigenvalues of a matrix