Matrices and Matrix Operations

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

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

Matrix arithmetic

• Sum ($A,B$ same size): $(A+B)_{ij} = a_{ij}+b_{ij}$
• Scalar: $(\alpha A)_{ij} = \alpha\,a_{ij}$
• Product (cols $A$ = rows $B$):

$$(AB)_{ij} = \sum_k a_{ik}\,b_{kj}$$

Important properties

• Multiplication is associative: $(AB)C = A(BC)$.
• Multiplication is not commutative: $AB \neq BA$ in general.
• The identity matrix $I_n$ satisfies $AI = IA = A$.
• The transpose: $(A^T)_{ij} = a_{ji}$.

Key insight: Every matrix represents a linear map between vector spaces.

Common pitfall: Matrix multiplication is not commutative: $AB \ne BA$ in general, because transformations compose in order — rotate-then-reflect differs from reflect-then-rotate. Never swap factors casually.