Mathematics · Ch 7 — Matrices

Matrix

7.2

Matrix

3.2 Matrix

The Idea Behind a Matrix

A matrix organises information in a rectangular arrangement. To record that Radha has 15 notebooks you could write $[15]$ ; if she also has 6 pens, $[15 \quad 6]$ , with the first number meaning notebooks and the second pens. Now take three people — Radha, Fauzia, and Simran — each with notebooks and pens:

	Notebooks	Pens
Radha	15	6
Fauzia	10	2
Simran	13	5

This information can be written in two equivalent rectangular arrangements:

\begin{bmatrix} 15 & 6 \\ 10 & 2 \\ 13 & 5 \end{bmatrix} \quad\text{or}\quad \begin{bmatrix} 15 & 10 & 13 \\ 6 & 2 & 5 \end{bmatrix}

In the first, each row is a person and each column an item; in the second, each row is an item and each column a person. Both are valid matrices.

Formal Definition

Definition 1: A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or entries of the matrix. Matrices are denoted by capital letters.

The word "ordered" is crucial — the position of each entry matters, so changing the order of rows or columns changes the matrix.

Examples of Matrices

A = \begin{bmatrix} 5 & -2 \\ 0 & 5 \\ 3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3.5 & -1 \\ \frac{2}{5} & \sqrt{3} & 5 + 7i \end{bmatrix}, \quad C = \begin{bmatrix} 3 & 1 & \sin x \\ \cos x & \tan x & x^2 + 3 \end{bmatrix}

Matrix $A$ contains only numbers; $B$ contains numbers, a fraction, a square root, and a complex number; $C$ contains functions of $x$ — trigonometric functions and a polynomial.

Rows and Columns

In any matrix, the horizontal lines of elements are called rows and the vertical lines are called columns. Matrix $A$ has 3 rows and 2 columns, $B$ has 3 rows and 3 columns, and $C$ has 2 rows and 3 columns.

Note

Always state the number of rows first, then columns. So $A$ is a $3 \times 2$ matrix (read "3 by 2"), $B$ is $3 \times 3$ , and $C$ is $2 \times 3$ .

General Notation

A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix. Its general form is

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = [a_{ij}]_{m \times n}$ …

Definition 1Matrix

Definition

A matrix is an ordered rectangular array of numbers or functions.

The numbers or functions inside the array are called its elements or entries.

Matrices are denoted by capital letters (e.g., $A$ , $B$ , $C$ ).

Ordered means the position of each entry matters — swapping rows or columns changes the matrix.
Rectangular means the entries are arranged in rows and columns, forming a rectangle (not a triangle or other shape).
The entries can be numbers (like $5$ , $-2$ , $3.5$ ) or functions (like $\sin x$ , $\cos t$ , $x^2$ ).

Intuition

Think of a matrix as a compact table that stores multiple pieces of related information in an organised grid — like a spreadsheet with rows and columns, where every cell has a fixed place.

Concrete Example

The matrix

$A = \begin{bmatrix} 15 & 6 \\ 10 & 2 \\ 13 & 5 \end{bmatrix}$ …

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