Matrices and Matrix Operations

Rectangular Arrays of Numbers

A matrix is a rectangular array of numbers arranged in rows and columns; an $m \times n$ matrix has $m$ rows and $n$ columns. Matrices compactly represent linear systems, transformations, graphs (adjacency matrices), and data tables — they are one of computing's workhorse structures.

The operations:

• Addition — add two matrices of the same shape 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 $B$. 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 a crucial quirk: 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) acts like the number 1: $AI = A$. A square matrix may have an inverse $A^{-1}$ with $AA^{-1} = I$, which "undoes" it — but not all matrices are invertible.

Why does this odd multiplication rule matter? Because a matrix represents a linear transformation, and multiplying matrices composes the transformations. Applying two transformations in sequence corresponds to one matrix product — which is why the "row-times-column" rule is exactly the right definition, and why order matters (doing A then B differs from B then A).

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 shapes must match ($A$'s columns $=$ $B$'s rows).