Matrices: Introduction & Operations
Matrices are rectangular arrays of numbers. They're the data structure of linear algebra — used in graphics, machine learning, physics simulations, and solving large systems of equations.
Key Concepts
Matrix Notation
An m×n matrix has m rows and n columns. Entry aᵢⱼ is in row i, column j. A 2×3 matrix has 2 rows and 3 columns. Vectors are 1×n (row) or n×1 (column) matrices.
Matrix Addition & Scalar Multiply
Add matrices of the same size by adding corresponding entries. Scalar multiply: k·A multiplies every entry by k. These operations are element-wise.
Matrix Multiplication
A (m×n) · B (n×p) = C (m×p). Entry cᵢⱼ = dot product of row i of A and column j of B. Note: AB ≠ BA in general. Non-commutative!
Special Matrices
Identity matrix I: 1s on diagonal, 0s elsewhere. AI = IA = A. Zero matrix: all entries 0. Transpose Aᵀ: swap rows and columns. Symmetric: A = Aᵀ.
Live Python Practice
Interactive Lab
Check Your Understanding
A 3×4 matrix has:
Matrix multiplication AB is defined when:
The identity matrix I satisfies: