Definition of a Matrix

DEFINITION 1.1.1 (Matrix)   A rectangular array of numbers is called a matrix.

We shall mostly be concerned with matrices having real numbers as entries.

The horizontal arrays of a matrix are called its ROWS and the vertical arrays are called its COLUMNS. A matrix having $m$ rows and $n$ columns is said to have the order $m \times n.$

A matrix $A$ of ORDER $m \times n$ can be represented in the following form:

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

where $a_{ij}$ is the entry at the intersection of the $i^{\mbox{th}}$ row and $j^{\mbox{th}}$ column.

In a more concise manner, we also denote the matrix $A$ by $[a_{ij}]$ by suppressing its order.

Remark 1.1.2   Some books also use $\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots &... ... & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$ to represent a matrix.

Let $A = \begin{bmatrix}1 & 3 & 7 \\ 4 & 5 & 6 \end{bmatrix}.$ Then $a_{11} = 1, \; a_{12} = 3, \; a_{13} = 7, \; a_{21} = 4, \; a_{22} = 5, \;$ and $\; a_{23} = 6.$

A matrix having only one column is called a COLUMN VECTOR; and a matrix with only one row is called a ROW VECTOR.

WHENEVER A VECTOR IS USED, IT SHOULD BE UNDERSTOOD FROM THE CONTEXT WHETHER IT IS A ROW VECTOR OR A COLUMN VECTOR.

DEFINITION 1.1.3 (Equality of two Matrices)   Two matrices $A=[a_{ij}]$ and $B= [b_{ij}]$ having the same order $m \times n$ are equal if $a_{ij} = b_{ij}$ for each $i =1, 2, \ldots, m$ and $j =1, 2, \ldots, n.$

In other words, two matrices are said to be equal if they have the same order and their corresponding entries are equal.

EXAMPLE 1.1.4   The linear system of equations $2 x + 3 y = 5$ and $3 x + 2 y = 5$ can be identified with the matrix $\begin{bmatrix}2 & 3 &: & 5 \\ 3 & 2 & : & 5\end{bmatrix}.$