Engineering / Mathematics — Modules, Vector Spaces and Matrices
Matrices: Basic Definitions and Properties
Matrices over a ring, their arithmetic, and the properties that survive when the base ring is not a field.
Executive summary
A matrix is a rectangular array of ring elements with addition and multiplication defined in the usual way. Multiplication is associative but not commutative, and the matrix ring is the standard example of a non-commutative ring.
Invertibility depends on the determinant being a unit, which over a field means merely non-zero and over the integers means plus or minus one.
Learning objectives
- Define matrix operations and verify associativity.
- State the invertibility criterion over a ring.
- Identify which properties require a field.
01Arithmetic
Matrix operations
For matrices over a commutative ring R: addition is entrywise, and
(AB)ᵢⱼ = Σ_k Aᵢₖ Bₖⱼ.
Multiplication is defined when the inner dimensions agree, is associative, and distributes over addition. It is not commutative even for square matrices, and this is the standard first example of a non-commutative ring.
| Property | Holds? |
|---|---|
| Associative | Yes |
| Distributive over addition | Yes |
| Commutative | No |
| Has identity | Yes, the identity matrix |
| Zero divisors | Yes, for n ≥ 2 |
02Invertibility
Invertibility criterion
A square matrix over a commutative ring R is invertible if and only if its determinant is a unit of R.
| Ring | Units | Invertible matrices |
|---|---|---|
| Field F | All non-zero elements | Determinant ≠ 0 |
| Z | ±1 | Determinant = ±1 |
| Z_n | Elements coprime to n | gcd(det, n) = 1 |
| F[X] | Non-zero constants | Determinant a non-zero constant |
03Matrices over finite fields
The case used throughout this collection is matrices over F_p, where the base is a field and all the familiar theory applies.
Index calculus
The relation matrix is over GF(2) for factoring and over Z_q for discrete logarithms. Both are fields, so standard elimination applies.
Berlekamp
Factorisation reduces to computing the kernel of a matrix over F_p, a pure rank computation.
Sparse solving
The matrices arising in sieve algorithms are enormous and sparse, so specialised methods replace dense elimination.
Counting invertible matrices over a finite field is a useful exercise with a clean answer: the number of invertible n × n matrices over F_q is the product of q^n − q^i for i from 0 to n−1, obtained by choosing each row outside the span of its predecessors.
|GL_n(F_q)| = ∏_{i=0}^{n−1} (q^n − q^i)04Frequently asked questions
Why are matrices excluded from the commutative ring theory here?
Because they are not commutative, and the whole development in this collection assumes commutativity. Matrices are handled as representations of module homomorphisms, where the relevant structure is the module rather than the matrix ring.
Does the determinant behave the same over any commutative ring?
Yes — the Leibniz formula and multiplicativity hold over any commutative ring. Only the interpretation of a non-zero determinant changes, since non-zero and unit are different conditions.
Are there zero divisors among matrices?
Yes, for size at least two. A non-zero matrix with a non-trivial kernel multiplied by a matrix whose columns lie in that kernel gives zero, and both factors are non-zero.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 316-320.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.