Determinants — Mathematics Knowledge System

Overview

The determinant is a scalar function det: M_{n x n}(F) -> F that encodes whether a square matrix is invertible and by what factor it scales volumes. For 2x2 matrices, det(A) = ad - bc; in general, it is the unique alternating multilinear form on the columns that is 1 on the identity. Determinants appear throughout linear algebra, multivariable calculus (Jacobians), differential geometry (volume forms), and algebra (characteristic polynomials).

Intuition

The determinant of a 2x2 matrix [[a,b],[c,d]] is ad - bc, which equals the signed area of the parallelogram spanned by the two columns [a,c] and [b,d]. For 3x3 matrices, it is the signed volume of the parallelepiped. A zero determinant means the matrix squishes space flat (the columns are linearly dependent and the matrix is singular). A determinant of 2 means the matrix doubles volumes.

Formal Definition

Definition

The determinant is characterised by three properties (Leibniz formula and Laplace expansion are consequences):

\det(I) = 1 \quad \text{(normalisation)}

Identity has determinant 1

\det \text{ is multilinear in each column}

Multilinearity

\det \text{ changes sign when two columns are swapped}

Alternating

\det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}

Leibniz formula (sum over permutations)

\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij}

Laplace expansion along row i

Notation

Notation	Meaning
$\det(A)$	Determinant of matrix A
$\|A\|$	Vertical bar notation for determinant
$M_{ij}$	Minor: determinant of submatrix with row i and column j removed
$C_{ij} = (-1)^{i+j}M_{ij}$	Cofactor

Theorems

Theorem 1: Multiplicativity

\det(AB) = \det(A)\det(B) \text{ for all } A, B \in M_{n\times n}(F)

Theorem 2: Invertibility Criterion

A \in M_{n\times n}(F) \text{ is invertible} \iff \det(A) \neq 0. \text{ In that case, } \det(A^{-1}) = 1/\det(A).

Theorem 3: Cramer's Rule

\text{For } Ax = b \text{ with } \det(A) \neq 0: \quad x_i = \frac{\det(A_i)}{\det(A)} \text{ where } A_i \text{ replaces column } i \text{ of } A \text{ with } b

Worked Examples

1
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
$\det\begin{pmatrix}3&1\\2&4\end{pmatrix} = 3\cdot4 - 1\cdot2 = 12 - 2 = 10$

✓ Answer

det = 10.

Practice Problems

Easyfree response

Show that if two rows of a matrix are identical, its determinant is 0.

Mediumfree response

If A is a 3x3 matrix with det(A) = 5, what is det(3A)?

Common Mistakes

Common Mistake

det(A + B) = det(A) + det(B)

The determinant is NOT additive. It is multilinear in the COLUMNS separately, not in the whole matrix. det(AB) = det(A)det(B), but det(A+B) has no simple formula.

Common Mistake

A positive determinant means the matrix is invertible

A NONZERO determinant (positive or negative) means invertible. det(A) > 0 additionally means A preserves orientation, but the sign doesn't affect invertibility.

Quiz

Historical Background

Determinants arose from the study of systems of linear equations in the 17th and 18th centuries. Leibniz (1693) and Cramer (1750) used determinant-like expressions. Vandermonde (1771) and Laplace (1772) developed expansion formulas. Cauchy gave the modern definition in 1812. Jacobi introduced the Jacobian determinant in 1841, linking determinants to change-of-variables in integration.

1693
Leibniz introduces the determinant in a letter about linear equations
Gottfried Leibniz
1750
Cramer publishes Cramer's rule for solving linear systems
Gabriel Cramer
1812
Cauchy gives the modern axiomatic treatment
Augustin-Louis Cauchy
1841
Jacobi introduces the Jacobian determinant
Carl Gustav Jacobi

Summary

The determinant det(A) is the unique alternating multilinear function on columns with det(I) = 1.
For 2x2: det[[a,b],[c,d]] = ad - bc (signed area of parallelogram of columns).
Invertibility criterion: A is invertible iff det(A) ≠ 0.
Multiplicativity: det(AB) = det(A)det(B).
Geometrically, |det(A)| is the volume scaling factor of the linear map A.

References

BookAxler, S. -- Linear Algebra Done Right (3rd ed., 2015), Chapter 10
BookStrang, G. -- Introduction to Linear Algebra (5th ed., 2016), Chapter 5
WebsiteWikipedia -- Determinant