Order of Matrix

The order of a matrix describes its dimensions in terms of the number of rows and columns. That is how many rows and columns. It is also represented as m × n, where m is the number of rows and n is the number of columns in the matrix. The diagram shows how the rows and columns are arranged in a matrix.
 

To determine the order of a matrix, first count the number of rows and the number of columns it has.

7
8
9

For example, 
Let A = 21 43 65 and B =   

Matrix A has 2 rows and 3 columns.
Matrix B has 3 rows and 1 column.

The two matrices are denoted as A and B. In a matrix, specific notation is used to identify the rows and columns.

A = [aij]
B = [bij]
i and j are indices that help locate the position of the elements in the matrix
i  denotes a row number.
j  denotes a column number.
For example,
Which is A = a21 = 2, a21 means the element in the 2nd row and 1st column of matrix A is 2.
Which is B = b11 = 7, b11 means the element in the 1st  row and 1st column of matrix B is 7.
 

The number of elements in a matrix is the total count of individual entries. It is found by multiplying the number of rows by the number of columns: number of elements = number of rows × number of columns = m × n, where m is the number of rows and n is the number of columns. For example, A = 53 97 1311. Matrix A has 2 rows and 3 columns, so it contains 6 elements.
 

 The order of the matrix indicates how many rows and columns it has. 

• Row matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Transpose Matrix

Row Matrix
A row matrix is a matrix with only one row and several columns. Its order is written as 1 × n.

Column Matrix:
A column matrix has only one column and multiple rows. So the order of the column matrix is m × 1.

Square Matrix
The square matrix has an equal number of rows and columns. So the order of a square matrix is n × n.

Rectangular Matrix
The matrix with different numbers of rows and columns is called a rectangular matrix. Its order is m × n, where m  n.

Transpose Matrix
The transpose matrix is the formation of the original matrix; its rows and columns are interchanged. If the matrix has an order which is m × n, then its transpose AT will be n × m.
 

The order of matrices plays a crucial role when performing operations like addition, subtraction, and multiplication.

Addition and Subtraction of Matrix
For two matrices to be added or subtracted, they must have the same order, which means the same number of rows and columns.

Multiplication Matrix
The multiplication of matrices has a condition on the order of the matrices. For number two matrices to be multiplied, the number of columns in the first matrix must be the same as the number of rows in the second matrix.
 

The order of a matrix has practical applications in daily life, especially in technology and data organization. Below are some examples of where it is used.

•  Photography and Graphics: The order of the matrix is used in digital images, which are made up of pixels arranged in rows and columns, just like elements of a matrix. 
• Seating Arrangements in Events or Classrooms: The order of the matrix is used to arrange the seats for events or classrooms. It can help to determine how many people can fit in the room. For example, if there are 10 rows of seats and 6 columns, the seating layout can be represented as a 10 × 6 matrix. This helps in organizing seat numbering, exits, and is useful for planning seating arrangements, like in auditoriums or exam centers.
•  Spreadsheet Data Entry: When we enter data into a spreadsheet, it is arranged in a grid of rows and columns, similar to the structure of a matrix.
•  Data Science and Machine Learning: In machine learning, datasets are often represented as matrices, with rows showing data samples and columns showing features or attributes.
•  Timetables and Scheduling: A school timetable can be represented as a matrix with rows indicating the days and columns indicating the class periods.