Matrices and Matrix Operations

Rectangular Arrays of Numbers

A matrix is a rectangular array of numbers in rows and columns; an $m \times n$ matrix has $m$ rows and $n$ columns. Matrices represent linear systems, transformations, and data — one of the central objects of linear algebra.

The operations:

• Addition — two matrices of the same shape add entry by entry.
• Scalar multiplication — multiply every entry by a number.
• Matrix multiplication — the important, non-obvious one. The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ (the inner dimensions match). Entry $(i, j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$. An $m \times n$ times an $n \times p$ matrix gives an $m \times p$ result.

Matrix multiplication has crucial quirks:

• It is not commutative — in general $AB \neq BA$, and often only one order is even defined.
• It is associative: $A(BC) = (AB)C$.
• The identity matrix $I$ (ones on the diagonal, zeros elsewhere) satisfies $AI = A = IA$, playing the role of 1.
• Some square matrices have an inverse $A^{-1}$ with $AA^{-1} = A^{-1}A = I$; those are the invertible (nonsingular) matrices.

The "row-times-column" rule looks strange until you know its purpose: a matrix represents a linear transformation, and matrix multiplication composes transformations. That is why the rule is exactly right, and why order matters — applying $A$ then $B$ differs from $B$ then $A$, just as function composition does.

Common pitfall: assuming matrix multiplication is commutative, or multiplying entry-by-entry like addition. In general $AB \neq BA$ (order matters, and often only one order is defined). And the product is not formed by multiplying corresponding entries — each result entry is a dot product of a row and a column, so the inner dimensions must match ($A$'s columns $=$ $B$'s rows).

Row i of A sweeping across column j of B, each pair multiplied and summed into the accent (i,j) entry of AB — the row-times-column rule.

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