2.1 Matrix basics

Matrices first appear in the course as a convenient way to record a system of linear equations. Very quickly, though, they become objects in their own right. To work rigorously later, you need to separate several ideas that beginners often blur together:

• what a matrix is;
• when two matrices are equal;
• which operations are defined before multiplication even enters the story;
• how a matrix records the data of a linear system.

This note lays down that language carefully.

Before you start

This section does not assume that you already know matrix theory. It does, however, assume a few earlier habits from algebra and the first systems note.

You should be able to:

• read variables such as $x_1$, $x_2$, and $x_3$ as distinct unknowns;
• recognize coefficients in a linear equation, including coefficients equal to 0, 1, and $-1$;
• keep an ordered tuple such as $(2,-1,4)$ in the correct order;
• do basic signed arithmetic entry by entry.

If any of these feel rusty, the right way to read this page is still not to memorize vocabulary first. Instead, pause at each worked example and ask: which row, which column, and which position is being used?

What a matrix is

If a matrix has m rows and n columns, we say it is an $m \times n$ matrix. The entries are usually real numbers in this course, although the formal definitions make sense over other number systems as well.

The point of the rectangular format is not decoration. A row records one line of data, a column records another line of data, and the position of an entry matters. Later, multiplication and row operations will depend on that position.

This is why a matrix should not be treated as a loose list of numbers. The entry in row 1, column 3 can play a completely different role from the same number placed in row 3, column 1. In a coefficient matrix, for example, columns are tied to the chosen order of the variables. If the variable order is $(x_1, x_2, x_3)$, then the second column records coefficients of $x_2$. If the order is changed to $(x_2, x_1, x_3)$, the displayed matrix must change as well.

Read the size before you touch the entries

The size of a matrix is written as $m × n$.

• m is the number of rows.
• n is the number of columns.

If $m = n$, the matrix is square.

Two matrices with different sizes are different kinds of objects. A $2 \times 3$ matrix and a $3 \times 2$ matrix are not even comparable entry by entry, because their row-column positions do not match.

Read entries one by one

If $A$ is a matrix, then $a_{ij}$ means the entry in row i and column j. That notation matters because it tells you exactly where a number lives inside the array.

The notation $a_{ij}$ is not optional bookkeeping. It is the language used in definitions such as matrix equality, matrix addition, and matrix multiplication.

For a first pass through this topic, practise reading before calculating. The interactive task below is deliberately simple: it trains the exact habits that later prevent mistakes in row reduction and matrix multiplication.

Matrix equality is entrywise

Two matrices are equal only when they have the same size and every corresponding entry matches.

This means that proving two matrices are equal is often an entry-by-entry argument.

Addition and scalar multiplication come first

Before matrix multiplication appears, there are two basic operations you should already read confidently.

The phrase "of the same size" is essential. Matrix addition is not defined for matrices of different sizes.

The zero matrix is the matrix whose entries are all 0. For each size it plays the role of the additive identity:

$$A + O = A.$$

A matrix records a linear system compactly

One reason matrices matter so early is that they package a linear system in a form that is easier to transform systematically.

Consider the system

$$\begin{aligned} x_1 + 2x_2 - x_3 &= 4, \\ 3x_1 - x_2 + 5x_3 &= 7. \end{aligned}$$

Its coefficient matrix is

$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & -1 & 5 \end{bmatrix},$$

its unknown vector is

$$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix},$$

and its constant vector is

$$b = \begin{bmatrix} 4 \\ 7 \end{bmatrix}.$$

So the whole system can be recorded as

$$Ax = b.$$

This compact form is not a shortcut that hides meaning. It gathers the same coefficients, variables, and constants into an object that later supports row operations, null-space language, and invertibility tests.

How to approach basic matrix questions

Most early matrix questions are not trying to surprise you. They are testing whether you can keep the bookkeeping straight. A reliable working order is:

1. identify the requested object: size, entry, row, column, coefficient matrix, constant vector, sum, or scalar multiple;
2. check whether the operation is defined before calculating;
3. keep row and column order fixed;
4. compute only the entries that the question asks for;
5. state the answer with its size when the object is a matrix or vector.

A short preview of multiplication

The next note explains matrix multiplication carefully. For now, you only need to see why rows and columns matter so much. Changing one row of a left matrix or one column of a right matrix changes exactly the output entries built from them.

Use the embedded figure as a preview of that row-by-column rule.

Common mistakes

Quick check

Exercise

Related notes

If you want to see how a system becomes a matrix, review 1.1 Equations and solution sets. For the next algebraic operation, continue to 3.1 Matrix multiplication and identity matrices.