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.
Key Formula
(AB)_ij = sum over k of a_ik * b_kj
LaTeX: (AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}
| Symbol | Meaning | Unit |
|---|---|---|
| A | m×n matrix | — |
| B | n×p matrix | — |
| a_{ik} | Entry in row i, column k of A | — |
| b_{kj} | Entry in row k, column j of B | — |
| (AB)_{ij} | Entry in row i, column j of product C | — |
Worked Example
Problem
Compute AB where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
Solution
Step 1: A is 2×2 and B is 2×2; result C is 2×2. Step 2: C₁₁ = (1)(5) + (2)(7) = 5 + 14 = 19. Step 3: C₁₂ = (1)(6) + (2)(8) = 6 + 16 = 22. Step 4: C₂₁ = (3)(5) + (4)(7) = 15 + 28 = 43. Step 5: C₂₂ = (3)(6) + (4)(8) = 18 + 32 = 50.
Answer
AB = [[19, 22], [43, 50]].
Properties of Matrix Multiplication
| Property | Statement | Holds? |
|---|---|---|
| Associativity | (AB)C = A(BC) | Yes |
| Commutativity | AB = BA | No (generally) |
| Distributivity | A(B + C) = AB + AC | Yes |
| Identity element | AI = IA = A | Yes |
| Zero product | AB = 0 implies A=0 or B=0 | No |
| Transpose of product | (AB)ᵀ = BᵀAᵀ | Yes |
Interactive Tools
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Related Terms
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.
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.
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/source) and Latin "multiplicatio" (a multiplying, from "multiplicare" — to multiply). Matrix multiplication as a distinct operation was introduced by Arthur Cayley in his 1858 "Memoir on the Theory of Matrices".