Determinants — Mathematics Knowledge System

Overview

The determinant is a scalar value computed from a square matrix that encodes geometric and algebraic properties. For 2x2: det([[a,b],[c,d]]) = ad - bc. For 3x3: cofactor expansion. A matrix is invertible iff its determinant is nonzero. The determinant gives the signed area/volume scaling factor of the linear transformation.

Intuition

The 2x2 determinant ad - bc measures the signed area of the parallelogram formed by vectors (a,c) and (b,d). If det = 0, the vectors are parallel and the parallelogram is flat -- no area -- which means the linear system either has no solution or infinitely many. Cramer's rule: each variable equals a determinant ratio.

Formal Definition

Definition

For 2x2: det[[a,b],[c,d]] = ad - bc. For 3x3 via cofactor expansion along row 1: det A = a_{11}*M_{11} - a_{12}*M_{12} + a_{13}*M_{13} where M_{ij} is the 2x2 minor. Properties: det(AB) = det(A)*det(B); det(A^T) = det(A); det(A^{-1}) = 1/det(A); swapping rows negates det; multiplying a row by k multiplies det by k.

\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc

2x2 determinant

\det A \neq 0 \iff A \text{ is invertible}

Invertibility criterion

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

Multiplicativity

Notation

Notation	Meaning
$\det(A)$	Determinant of matrix A
$\|A\|$	Alternative determinant notation

Theorems

Theorem 1: Cramer's Rule

For Ax = b with det(A) not 0, the unique solution has x_i = det(A_i)/det(A) where A_i is A with column i replaced by b. Practical for small systems; inefficient for large ones.

Worked Examples

1
2x2 formula: ad - bc.
$\det = 3 \cdot 4 - (-2) \cdot 1 = 12 + 2 = 14$

✓ Answer

14.

Practice Problems

Mediumfill in blank

Use Cramer's rule to solve: 2x + y = 5, x - 3y = -4.

Common Mistakes

Common Mistake

Using ad - bc for a 3x3 matrix.

The formula ad - bc only works for 2x2 matrices. For 3x3, use cofactor expansion along a row or column.

Quiz

If det(A) = 0, then the matrix A:

Has a unique inverseIs not invertibleHas determinant 1Is the identity

Historical Background

Determinants were introduced by Leibniz (1693) and independently by Seki Takakazu in Japan around the same time. Cramer's rule (1750) gave explicit formulas for linear systems. Cauchy (1815) established many properties, and the theory was systematized by Jacobi (1841). The geometric interpretation as area/volume scaling was recognized in the 19th century.

1693
Leibniz introduces determinants for solving linear systems
Gottfried Wilhelm Leibniz
1750
Cramer publishes his rule for solving linear systems using determinants
Gabriel Cramer

Summary

2x2 det: ad - bc. 3x3: cofactor expansion.
det(A) = 0 iff A is singular (not invertible).
det(AB) = det(A)*det(B). det(A^T) = det(A).
Geometric meaning: |det(A)| = area/volume scaling factor of the linear map.

References

BookLarson, R. Algebra and Trigonometry. 9th ed. Brooks/Cole, 2013.