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Matrices in computer applications Matrices in computer applications One of the most important usages of matrices in computer side applications are encryption of message codes. Matrices and their inverse matrices are used for a programmer for coding or encrypting a message. A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving. Hence with the help of matrices, those equations are solved. With these encryptions only, internet functions are working and even banks could work with transmission of sensitive and private data.
Introduction to matrices

A rectangular arrangement of numbers in the form of horizontal and vertical lines is called a matrix (plural form: "matrices").
The horizontal lines are called rows and the vertical lines are called columns of a matrix.
Numbers that appear in the rows and columns of a matrix are called its elements or its entries.
All the elements are enclosed in square brackets [ ] or parentheses ( ).
Ex: The arrangement of the numbers 10, 11, 12, 20, 22, 24, 30, 33, 36 as below.

In the above matrix, 11, 22, 33, 12, 24, 36, 10, 20 and 30 are called its elements.
The number of rows and columns that a matrix has is called its dimension or its order.
As a convention, rows are listed first and columns are listed second. Thus, we would say that the dimension (or order) of the above matrix is 3 × 3 (read as "3 by 3"), meaning that it has 3 rows and 3 columns.

Example

Matrices in robotics and automation Matrices in robotics and automation In robotics and automation, matrices are the base elements for the robot movements. The movements of the robots are programmed with the calculation of rows and columns of matrices. The inputs for controlling robots are given based on the calculations from matrices.
Elements of a rectangular matrix
Q: The order of a matrix A is 3 × 5. List its elements in each row in general form.
Sol: The matrix has 3 rows and 5 columns i.e, m = 3 and n = 5
The total number of elements = m.n = 15
The general form of an element = aij
The max. value of i = m = 3
The max. value of j = n = 5
∴ i takes the values 1, 2 & 3
and j takes the values 1, 2, 3, 4 & 5
For i = 1, the elements in first row are:
a11, a12, a13, a14, a15
For i = 2, the elements in second row are:
a21, a22, a23, a24, a25
For i = 3, the elements in third row are:
a31, a32, a33, a34, a35
Matrix Notation

A matrix is usually represented by an upper–case letter such as A, B, P, Y etc.
Its elements are represented by corresponding lower–case letters with two subscript indices such as eg: a11, b21, p43, y23 etc.
In general, aij represents an element of the matrix A occurring in ith row and jth column.

In the 2 × 2 matrix
a11 represents the element in the first row and first column;
a12 represents the element in the first row and second column;
a21 represents the element in the second row and first column and
a22 represents the element in the second row and second column.

General form of a matrix:

In general, a set of 'mn' elements can be arranged as a matrix having 'm' rows and 'n' columns as below:

Or in short,

    A = [aij]m × n

In the above matrix, aij represents an element of the matrix A occurring in ith row and jth column.
The maximum value of 'i' can be 'm' i.e, the number of rows and
the maximum value of 'j' can be 'n' i.e, the number of columns.

In general, any matrix of the order m × n is called a rectangular matrix.

Types of Matrices - Square Matrix

A matrix in which number of rows is equal to number of columns is called a square matrix.
Thus, an m × n matrix for which m = n is called a square matrix of order ‘n’.
In the below figure, the matrices A and B are square matrices of order 3 and 2 respectively.

Consider a square matrix, A = [aij]n × n
All the elements aij for which i = j, are called the diagonal elements of the matrix.
i.e, the elements a11, a22, a33, .... and ann
The line along which they lie is called the principal diagonal of the matrix.
For the 3 × 3 matrix A above, 1, 5, 9 are the diagonal elements which also constitute the principal diagonal of the matrix A (as shown by the line in red below).
Similarly for the 2 × 2 matrix B above, the elements 2, 7 constitute its principal diagonal.

Note: A matrix having a single element, such as [a11], is also a square matrix because i = j = 1.

Trace of a matrix

If A is a square matrix, then the sum of the elements in the principal diagonal of A is called trace of A. It is denoted by tra A.