Matrix Addition

Matrix Basics

A matrix is a rectangular array of numbers representing data like sales, costs, or transitions.

Dimension is written as $m \times n$ ($m$ rows, $n$ columns). The entry $a_{ij}$ sits in row $i$, column $j$. Order is always rows first.

$$A_{2\times 3} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}$$

Common pitfall: Reading $a_{ij}$ column-first. Mixing this up silently transposes every calculation that follows.

Addition & Scaling

Matrix addition works entry by entry, and only between matrices of the same dimension: $(A+B)_{ij} = a_{ij} + b_{ij}$. Adding a $2\times 3$ to a $3\times 2$ is as meaningless as adding a price list to a staff roster.

Scalar multiplication rescales every entry at once: $(kA)_{ij} = k\,a_{ij}$. A 10% price rise is $1.1\,P$.

Properties: Addition is commutative ($A+B = B+A$), associative, has a zero matrix, and scalars distribute: $k(A+B) = kA + kB$.

The payoff is compression: one symbol carries a whole spreadsheet, and one equation performs hundreds of operations in a single conceptual move.