Matrices and Matrix Operations

Matrix Basics and Addition

A matrix is a rectangular array of numbers organized in rows and columns. An $m \times n$ matrix has $m$ rows and $n$ columns. They represent linear systems, transformations, and data.

Common pitfall: Never add matrices entry-by-entry if their shapes differ, and never multiply matrices entry-by-entry like addition. Matrix multiplication follows the strict row-times-column rule.

Matrix Multiplication

The product $AB$ is defined only when the inner dimensions match: the number of columns of $A$ equals the number of rows of $B$. An $m \times n$ times an $n \times p$ matrix yields an $m \times p$ result.

Entry $(i,j)$ is the dot product of row $i$ of $A$ and column $j$ of $B$:

$$(AB)_{ij} = \sum_k A_{ik} B_{kj}$$

Matrix multiplication is not commutative ($AB \neq BA$), but it is associative ($A(BC) = (AB)C$). The identity matrix $I$ has ones on the diagonal and satisfies $AI = IA = A$. Square matrices with an inverse $A^{-1}$ where $AA^{-1} = A^{-1}A = I$ are invertible (nonsingular).