Matrix Multiplication

Compute \(C=AB\) and inspect each result cell's dot product.

A 2x3 x B 3x2 = C 2x2 8 multiply-add terms

Matrix A

2 rows x 3 columns

Matrix B

3 rows x 2 columns

Operation

A has 3 columns and B has 3 rows. The matrices are compatible.

Ready.

Result C

2 rows x 2 columns

Dot product preview Run multiplication to see how a result cell is formed.

Geometric composition

See the same square at every step: B acts first, then A acts on B's result.

Transformed unit square First basis vector e₁ Second basis vector e₂

Choose the 2D geometry preset to visualize two 2×2 transformations.

How to read the product

Each output cell is a dot product: one row from A paired with one column from B. Select a result cell after multiplying to inspect its arithmetic.

Output shape: —Dot products: —Terms per cell: —Scalar operations: —

Mathematical Background

Matrix Multiplication - Introduction

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a third matrix. Unlike scalar multiplication, matrix multiplication is not element-wise; instead, it involves a systematic combination of rows and columns.

This operation is essential for representing compositions of linear transformations, solving systems of equations, and modeling complex relationships in science, engineering, and computer graphics.

Mathematical Definition

Given two matrices A (m×n) and B (n×p), their product C = AB is an (m×p) matrix where each element is:

\[C_{ij}=\sum_{k=1}^{n}A_{ik}B_{kj}\]

In other words, element C[i,j] is computed by taking the dot product of the i-th row of matrix A and the j-th column of matrix B.

Dimension Compatibility: For AB to be defined, columns in A must equal rows in B.

Properties of Matrix Multiplication

Matrix multiplication has several important properties:

  • Associativity: (AB)C = A(BC)
  • Distributivity: A(B + C) = AB + AC
  • Non-Commutativity: AB ≠ BA in general - order matters!
  • Identity: AI = IA = A for identity matrix I
  • Transpose: (AB)T = BTAT

Geometric Interpretation

Matrix multiplication represents composition of linear transformations:

  • Each matrix represents a transformation (rotation, scaling, etc.)
  • Product AB means: apply B first, then A
  • This explains why AB ≠ BA
  • Used in 3D graphics for transformations

Applications

  • Computer Graphics: 3D transformations and rendering
  • Neural Networks: Deep learning computations
  • Quantum Mechanics: State evolution
  • Markov Chains: Probability transitions
  • Robotics: Kinematics calculations
  • Graph Theory: Path counting

Computational Complexity

Standard algorithm for n×n matrices: O(n³) operations.

Optimizations:

  • Strassen's Algorithm: O(n2.807)
  • GPU acceleration for parallel processing
  • Sparse matrix methods

Common Mistakes

  • Dimension mismatch (columns of A must equal rows of B)
  • Assuming AB = BA (it's not commutative!)
  • Confusing with element-wise multiplication

Tips: Always check dimensions first: (m×n) × (n×p) → (m×p)