Introduction to Matrices

Matrix

A matrix (plural: matrices) is a rectangular arrangement of numbers, symbols, or expressions organized into rows and columns.

For example,

$$A= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$

• It has 2 rows.
• It has 3 columns.
• So it is called a $2 \times 3$ matrix $rows × columns$.

General form

A matrix with $m$ rows and $n$ columns is written as

$$A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

where:

• $m$ = number of rows
• $n$ = number of columns
• $a_{ij}$ is the element in the $i$-th row and $j$-th column.

Example

Consider

$$B= \begin{bmatrix} 7 & 8\\ 9 & 10\\ 11 & 12 \end{bmatrix}$$

• Order: $3 \times 2$
• $b_{11}$=7
• $b_{23}=,$does not exist (there are only 2 columns)
• $b_{32}=12$

Why are matrices useful?

Matrices are used to:

• Solve systems of linear equations.
• Represent and transform geometric objects (e.g., rotate, scale, translate points in computer graphics).
• Organize and analyze data.
• Represent networks and graphs.
• Perform computations in machine learning and artificial intelligence.

Common types of matrices

Type	Example
Row matrix	$\begin{bmatrix}1 & 2 & 3\end{bmatrix}$
Column matrix	$\begin{bmatrix}1\\2\\3\end{bmatrix}$
Square matrix	$\begin{bmatrix}1&2\\3&4\end{bmatrix}$
Zero matrix	$\begin{bmatrix}0&0\\0&0\end{bmatrix}$
Identity matrix	$\begin{bmatrix}1&0\\0&1\end{bmatrix}$
Diagonal matrix	$\begin{bmatrix}2&0\\0&5\end{bmatrix}$

In simple terms, a matrix is like a table of values. Just as a spreadsheet stores data in rows and columns, a matrix stores numbers in rows and columns, but with mathematical rules that allow you to perform operations such as addition, multiplication, finding inverses, and solving equations.