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Definition of a 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 × n.

A matrix A of order m × n can be represented in the following form:

where aij is the entry at the intersection of the ith row and jth column.

In a more concise manner, we also denote the matrix A by [aij] by suppressing its order.  
Note: Some books also use
 
to represent a matrix.
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.

Here are a couple of examples of different types of matrices:
Symmetric:


Diagonal


Upper Triangular


Lower Triangular


Zero


Identity


Example 1:
Let , list out the aij’s values in A.
Solution: 
a11 = 1, a12 = 3, a13 = 6
a21 = 2, a22 = 3, a23 = 7
a31 = 4, a32 = 4, a33 = 0

Example 2:
State the aij’s values in .
Solution:
a11 = 9, a12 = 8, a13 = 7
a21 = 6, a22 = 5, a23 = 4
a31 = 3, a32 = 2, a33 = 1
a41 = 4, a42 = 6, a43 = 8
Example 3: Provide two examples of column and row matrices each.
Solution:
We know that, a matrix having only one column is called a column vector or column matrix and a matrix with only one row is called a row vector or row column.


Example 4: Is the following matrix classified under the category of matrices?

Solution:
A is not a matrix because column three or we can say row three is incomplete.

Example 5: What is the order of the following matrix?

Solution:
We know that a matrix having m rows and n columns is said to have the order m × n, therefore the order of A is 4 × 3.
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