Matrices and Matrix Operations

Matrix Basics

A matrix is a rectangular array of numbers arranged in rows and columns. An $m \times n$ matrix has $m$ rows and $n$ columns.

$$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$

Every matrix represents a linear map between vector spaces. Transformations compose in order, so matrix multiplication depends strictly on factor sequence.

Rules and Pitfalls

Matrix arithmetic follows strict algebraic laws. The identity matrix $I_n$ acts as a neutral element such that $AI = IA = A$.

The transpose operation flips a matrix across its main diagonal, defined by $(A^T)_{ij} = a_{ji}$. Multiplication is associative, meaning $(AB)C = A(BC)$.

Common pitfall: Matrix multiplication is not commutative ($AB \neq BA$). Rotate-then-reflect differs from reflect-then-rotate. Never swap factors casually.