Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
Key Formula
A = [[a11, a12, ..., a1n], [a21, a22, ..., a2n], ..., [am1, am2, ..., amn]]
LaTeX: A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}
| Symbol | Meaning | Unit |
|---|---|---|
| a_{ij} | Entry in row i, column j | — |
| m | Number of rows | — |
| n | Number of columns | — |
Worked Example
Problem
Given A = [[2, 1], [5, 3]] and b = [[8], [20]], solve the system Ax = b.
Solution
Step 1: Write augmented matrix [A | b] = [[2, 1, | 8], [5, 3, | 20]]. Step 2: R2 ← R2 − (5/2)R1: [[2, 1, | 8], [0, 1/2, | 0]]. Step 3: From row 2: (1/2)x₂ = 0 → x₂ = 0. Step 4: From row 1: 2x₁ + 0 = 8 → x₁ = 4.
Answer
x₁ = 4, x₂ = 0; solution vector x = (4, 0).
Common Types of Matrices
| Type | Definition | Example Use |
|---|---|---|
| Square matrix | m = n (equal rows and columns) | Linear transformations in ℝⁿ |
| Identity matrix I | Diagonal entries = 1, rest = 0 | Neutral element for multiplication |
| Symmetric matrix | A = Aᵀ (equals its transpose) | Covariance matrices in statistics |
| Diagonal matrix | Non-zero entries only on diagonal | Scaling transformations |
| Orthogonal matrix | AᵀA = I (columns are orthonormal) | Rotation and reflection matrices |
| Zero matrix | All entries = 0 | Additive identity |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Determinant
The determinant is a scalar value computed from a square matrix that encodes whether the linear transformation represented by the matrix is invertible, scaling the volume by a factor equal to the absolute value of the determinant and changing orientation if the determinant is negative. For a 2×2 matrix, det(A) = ad − bc; for larger matrices, it is computed recursively via cofactor expansion or row reduction. The determinant is zero if and only if the matrix is singular (non-invertible), making it a critical tool in solving linear systems, computing eigenvalues, and computing cross products in geometry.
Matrix Multiplication
Matrix multiplication is a binary operation that takes an m×n matrix A and an n×p matrix B and produces an m×p matrix C, where each entry Cᵢⱼ is the dot product of the i-th row of A with the j-th column of B. Unlike scalar multiplication, matrix multiplication is not commutative (AB ≠ BA in general) but is associative and distributive. It represents the composition of linear transformations and is the central computational operation in linear algebra, computer graphics, neural networks, and the numerical solution of differential equations.
Eigenvalue
An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v (its eigenvector) satisfying Av = λv, meaning the matrix only scales the vector without changing its direction. Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, and they encode fundamental properties of the linear transformation including stability, principal stresses, and oscillation frequencies. Applications span quantum mechanics (energy levels), structural engineering (natural frequencies), principal component analysis (data variance), and PageRank (web link importance).
From Latin "matrix" (womb, origin, source), used by James Joseph Sylvester in 1850 to describe an array from which determinants can be formed. Sylvester chose the term because the matrix "gives birth" to the determinant.