Mathematics.

matrix algebra

Determinant

Linear Algebra30 minDifficulty5 out of 10

You should know: matrices

Overview

The determinant is a scalar-valued function of a square matrix's entries, denoted det(A), det A, or |A|. It characterizes key properties of the matrix and the linear map it represents: the determinant is nonzero exactly when the matrix is invertible (equivalently, when the linear map is an isomorphism). A matrix with determinant zero is called singular and has no inverse. Geometrically, |det(A)| is the factor by which A scales area (2×2) or volume (3×3 and higher).

Intuition

Picture the unit square (or unit cube in 3D) transformed by a matrix A. The determinant tells you the signed scaling factor of that area or volume after the transformation: |det(A)| = 2 means areas double, det(A) = 0 means the transformation flattens everything onto a lower-dimensional subspace (collapsing area/volume to zero — irreversible, hence 'singular'), and a negative determinant signals the transformation flips orientation (like a reflection).

Interactive Graph

The determinant is the area-scaling factor of this transformation

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Formal Definition

Definition

For a 2×2 matrix, the determinant is defined directly. For a 3×3 matrix, the rule of Sarrus gives an explicit formula (valid only for 3×3):

det⁡(abcd)=ad−bc\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

The 2×2 determinant formula

2×2 case
det⁡(abcdefghi)=aei+bfg+cdh−ceg−bdi−afh\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh

Sum of forward diagonal products minus sum of backward diagonal products

3×3 case (rule of Sarrus)

Notation

NotationMeaning
det⁡(A)\det(A)The determinant of square matrix A
∣A∣|A|Alternative notation for det(A), using vertical bars around the matrix entries

Properties

Invertibility criterion

A is invertible  ⟺  det⁡(A)≠0A \text{ is invertible} \iff \det(A) \neq 0

Determinant of a product

det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B)

Determinant of the transpose

det⁡(AT)=det⁡(A)\det(A^{\mathsf T}) = \det(A)

Repeated row/column gives zero

∣abab∣=ab−ba=0\begin{vmatrix} a & b \\ a & b \end{vmatrix} = ab - ba = 0

Example: Any matrix with two identical rows or columns has determinant 0

Multilinearity in each row

∣ab+b′cd+d′∣=∣abcd∣+∣ab′cd′∣\begin{vmatrix} a & b+b' \\ c & d+d' \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} a & b' \\ c & d' \end{vmatrix}

Example: The determinant is linear in each row (and column) separately

Determinant of identity

det⁡(In)=1\det(I_n) = 1

Scaling a row scales the determinant

det⁡(cA)=cndet⁡(A)\det(cA) = c^n \det(A)

Condition: For an n×n matrix A, scaling ALL entries by c

Applications

Determinants test whether a system of linear equations has a unique solution (Cramer's rule) and whether a set of vectors is linearly independent.

Worked Examples

  1. Apply the 2×2 formula ad − bc.

    det⁡=(3)(−4)−(7)(1)=−12−7=−19\det = (3)(-4) - (7)(1) = -12 - 7 = -19

Answer: -19

Practice Problems

Difficulty 3/10

Compute det([[2, 5], [1, 3]]).

Difficulty 4/10

If det(A) = 0 for a square matrix A, what can you conclude?

Difficulty 5/10

A land parcel has corners at (0,0), (6,0), (6,4), (0,4) but was surveyed as a parallelogram spanned by vectors u = (6, 0) and v = (2, 4). Use the determinant to find its area.

Difficulty 6/10

Why does a structural engineer care that the global stiffness matrix K becomes SINGULAR (det K = 0) before supports are applied?

Common Mistakes

Common Mistake

Applying the rule of Sarrus (diagonal shortcut) to matrices larger than 3×3.

The Sarrus diagonal trick is specific to 3×3 matrices. For 4×4 and larger, use cofactor (Laplace) expansion along a row/column or row-reduce to triangular form instead.

Common Mistake

Assuming det(A+B) = det(A) + det(B).

The determinant is NOT additive over matrix addition — only multiplicative over matrix multiplication (det(AB)=det(A)det(B)) and linear in each individual row/column, not the whole matrix sum.

Quiz

A square matrix A has det(A) = 0. This means A is:
Geometrically, the absolute value of a 2×2 determinant equals:
Which identity is TRUE for determinants?

Summary

  • det(A) is a scalar computed from a square matrix's entries: ad−bc for 2×2, and the Sarrus diagonal formula for 3×3.
  • A matrix is invertible if and only if its determinant is nonzero; det(A)=0 means A is singular.
  • det(AB) = det(A)det(B), and det(Aᵀ) = det(A).
  • |det(A)| measures the area/volume scaling factor of the linear transformation A represents; a negative sign indicates orientation reversal.
  • For 4×4 and larger matrices, cofactor expansion or row reduction replaces the Sarrus shortcut.

References

  1. WebsiteWikipedia — Determinant
  2. BookStrang, G. Introduction to Linear Algebra, 5th ed. Ch. 5.