Introduction

Matrices are one of the most useful tools in mathematics. You can think of a matrix as nothing more than a structured way to store numbers—a data table.

If you know basic algebra, you already have everything you need to begin.

What Is a Matrix?

A matrix is a rectangular grid of numbers. For example: $$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$ Key ideas:

• A matrix has rows (horizontal) and columns (vertical).
• The above matrix has 2 rows and 3 columns.
• We call this a $2 \times 3$ matrix (“2 by 3”).
• Each number inside the matrix is called an entry.

You can think of a matrix as:

• A spreadsheet
• A small table of data
• A way to organize information neatly

Matrices as Data Structures

Matrices are not just mathematical objects—they are containers.

They can store:

• Measurements
• Coordinates
• Relationships
• Transformations
• Any structured numerical data

Examples of matrices as data structures:

• A matrix storing temperatures over several days
• A matrix storing pixel values in an image
• A matrix storing student test scores

Matrices help us:

• Organize data
• Perform operations on entire sets of numbers at once
• Represent systems compactly

Matrix Notation and Terminology

Some gentle notation:

• $A$ usually denotes a matrix.
• $A_{ij}$ means “the entry in row $i$, column $j$.”
• A matrix with $m$ rows and $n$ columns is an $m \times n$ matrix.

Example:

If $$A = \begin{pmatrix} 7 & 8 \\ 9 & 10 \end{pmatrix}$$ Then:

• $A_{11} = 7$
• $A_{12} = 8$
• $A_{21} = 9$
• $A_{22} = 10$

Matrix Addition

You can only add matrices of the same size.

Add them entry‑by‑entry: $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$

Scalar Multiplication

Multiply every entry by the same number: $$3 \begin{pmatrix} 2 & 1 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ 0 & -3 \end{pmatrix}$$ These operations mirror how you might scale or combine data tables.

Why Matrices Matter

Matrices are everywhere:

• Computer graphics
• Machine learning
• Physics
• Economics
• Engineering
• Statistics

They allow us to:

• Represent large sets of equations
• Transform geometric objects
• Store and manipulate data efficiently

Even at a beginner level, matrices give you a powerful new way to think about information.

Calculator

Exercises

1. Identify the size (rows and columns) of the matrix $$\begin{pmatrix} 3 & 1 & 4 \\ 1 & 5 & 9 \end{pmatrix}$$

   Solution

   The matrix has 2 rows and 3 columns, so it is a $2 \times 3$ matrix.
2. Compute $$2\begin{pmatrix} 1 & 0 \\ -1 & 3 \end{pmatrix}$$

   Solution

   $$2\begin{pmatrix} 1 & 0 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ -2 & 6 \end{pmatrix}$$
3. Add the matrices $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$$

   Solution

   $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}$$
4. True or false: You can add a $2 \times 3$ matrix to a $3 \times 2$ matrix.

   Solution

   False. Matrix addition requires both matrices to have the same size.
5. Find the entry in row 2, column 3 of $$\begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix}$$

   Solution

   The entry in row 2, column 3 is 8.
6. Multiply the matrix $$\begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix}$$ by $-3$.

   Solution

   $$-3\begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} -6 & 3 \\ 0 & -12 \end{pmatrix}$$
7. Describe in words what a matrix is used for.

   Solution

   A matrix is a structured table of numbers used to organize, store, and manipulate data.