Foundations — Matrix operations — addition, subtraction (conditions)

Before you can add matrices, you must be fluent in the language the parent page speaks. Below is every symbol and idea it uses, built from absolute zero, each one resting on the one before it.

1. What a matrix even is — the grid

Look at the figure. A row runs left-to-right (horizontal). A column runs top-to-bottom (vertical). The whole thing is one object we give a single capital-letter name, like $A$.

Why the topic needs it: you cannot "add two things" until you know what one thing is. The grid is the object; addition is an action on grids.

See Matrix Notation and Terminology for the full vocabulary tour.

2. Rows, columns, and the "address" of a number

Every number inside the grid sits in exactly one box, and that box has an address: which row, then which column.

Why the topic needs it: the parent page's whole argument — "add entries that occupy the same logical slot" — is meaningless unless "slot" (= address) is crystal clear. Two numbers can only be added together if they share an address.

3. The symbol $a_{ij}$ — reading a subscript

Writing out "the number in row 2, column 3 of matrix $A$" every time is exhausting. Mathematicians shrink it to $a_{23}$.

Why the topic needs it: the addition rule is literally written as $c_{ij}=a_{ij}+b_{ij}$. Without knowing that $i$ is the row and $j$ is the column, that line is gibberish.

4. Order / dimensions — the shape $m\times n$

The shape of a matrix is described by two counts.

Why the topic needs it: the single condition for adding matrices is "orders must be identical". Order is the yardstick that decides whether two grids can meet.

5. The bracket-notation shorthand $A=[a_{ij}{]}_{m\times n}$

The parent page writes the whole grid compactly as $A=[a_{ij}{]}_{m\times n}$. Now every piece is earned:

Why the topic needs it: this is the compressed sentence the definition and derivation sections rely on. It lets us talk about all entries at once by talking about one generic entry $a_{ij}$.

6. The $+$ and $-$ signs — and "element-wise"

You already know $+$ and $-$ for ordinary numbers. The new phrase is element-wise.

$c_{ij}=a_{ij}+b_{ij}\text{for every address }(i,j)$

Why the topic needs it: "element-wise" is the engine of both addition and subtraction. It's why matching shapes matters — every box in one grid must find its partner box in the other.

7. Special grids you'll meet: the zero matrix $O$

Why the topic needs it: the parent page's identity property says $A+O=A$ — adding zeros changes nothing, box by box. Full detail lives in Zero Matrix and Identity Matrix.

8. The negative of a matrix, $-A$

Why the topic needs it: the parent page defines subtraction as $A-B=A+(-B)$ and the additive-inverse property as $A+(-A)=O$. Both need $-A$ defined first. Flipping signs of every entry is exactly Scalar Multiplication of Matrices with the scalar $-1$.

9. Two ideas you will hear named but NOT need to compute here

These appear in the parent page's contrast notes, so recognise them — they are neighbours, not tools you use to add.

• Transpose $B^T$ — flipping a grid so its rows become columns (turns $m\times n$ into $n\times m$). The parent warns $A+B^T\ne A+B$. See Transpose of a Matrix.
• Matrix multiplication — a different operation with a different shape rule ($m\times n$ times $n\times p$). Do not confuse its rule with the addition rule. See Matrix Multiplication.

Prerequisite map

Read it top-down: the grid gives boxes, boxes give addresses, addresses give the symbol $a_{ij}$, shapes give the order, and together with the same-shape rule they power element-wise addition — the parent topic.

Equipment checklist

Test yourself — say the answer aloud before revealing.

Connections

• Matrix operations — addition, subtraction (conditions) — the parent this page prepares you for
• Matrix Notation and Terminology — the source of order, entries, and $a_{ij}$
• Zero Matrix and Identity Matrix — where $O$ comes from
• Scalar Multiplication of Matrices — $-A$ is scalar $-1$ times $A$
• Transpose of a Matrix — the neighbour $B^T$ that is not the same as $B$
• Matrix Multiplication — a different operation with a different shape rule
• System of Linear Equations — where these grids come from in the first place