Determinant
Formulas
| Property | Statement |
|---|---|
| 2×2 | ad−bc |
| 3×3 | Cofactor expansion |
| Geometric | |det A| = volume scaling factor |
| Invertible | det A ≠ 0 iff A invertible |
Examples
Example 1. det([[1,2],[3,4]]).
Solution. 1·4−2·3=−2.
In Depth
The determinant of an \(n\times n\) matrix is a scalar that encodes the signed volume scaling factor of the corresponding linear transformation. If \(|\det A|>1\), the transformation expands volumes; if \(|\det A|<1\), it contracts them; if \(\det A=0\), the transformation collapses the space to lower dimension (the matrix is singular).
The sign of \(\det A\) indicates orientation: positive means the transformation preserves orientation (like a rotation); negative means it reverses orientation (like a reflection). This is why \(\det(AB)=\det A\cdot\det B\) — composing transformations multiplies their volume scaling factors.
Cramer's rule expresses the solution of \(Ax=b\) as \(x_i=\det(A_i)/\det(A)\) where \(A_i\) replaces the \(i\)th column of \(A\) with \(b\). While theoretically elegant, it is computationally inefficient for large systems — Gaussian elimination is far faster in practice.
Further Reading & Context
The study of determinant connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.
Deep Dive: Determinant
This lesson extends core ideas for determinant with rigorous reasoning, edge-case checks, and application framing in linear algebra.
Practice Set
Practice. Derive one main result on this page and validate with a numeric or geometric check.
Goal. Confirm assumptions, transformation steps, and final interpretation.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Last editorial review: 2026-04-14.