Matrix Addition

Tables That Do Arithmetic

A matrix is a rectangular array of numbers — and in business, matrices are everywhere before you notice them: a sales table (rows = products, columns = branches), a cost breakdown, a transition table of customers between plans.

Notation: a matrix $A$ of dimension $m \times n$ has $m$ rows and $n$ columns; the entry $a_{ij}$ sits in row $i$, column $j$. Order matters and is read rows first:

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

Addition works entry by entry — and only between matrices of the same dimension:

$$(A+B)_{ij} = a_{ij} + b_{ij}$$

If January's sales matrix and February's sales matrix have the same layout, $A + B$ is the two-month total, cell by cell. 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% across-the-board price rise is one scalar: $1.1\,P$. Budgeting next year as 120% of this year: $1.2\,B$.

The algebra is friendly: addition is commutative ($A+B = B+A$), associative, has a zero matrix, and scalars distribute: $k(A+B) = kA + kB$.

The payoff of this notation is compression: one symbol carries a whole spreadsheet, and one equation — $T = A + B$ or $P' = 1.1P$ — performs hundreds of cell operations in a single conceptual move. That compression is what will let us solve entire systems of equations at once in the coming lessons.

Matrix operations, first rules

Common pitfall: Reading $a_{ij}$ column-first. The convention is rows first, always: $a_{23}$ lives in row 2, column 3 — mixing this up silently transposes every calculation that follows.