User:Ananias Amadhila

'''Demystifying Matrices: A Step-by-Step Guide to Mastering Matrices for Beginners

Introduction to Matrices'''

The matrices are a two-dimensional set of numbers or symbols distributed in a rectangular shape in vertical and horizontal lines so that their elements are arranged in rows and columns. They are useful for describing systems of linear or differential equations, as well as representing a linear application. A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to get some idea of what matrices are all about, we will look at the following example.

Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 2014 and 2015 is listed in the table below. 2014 2015 CHAIRS TABLES CHAIRS TABLES SAN JOSE 30 18 36 20 HAYWARD 20 12 24 18 OAKLAND 16 10 20 12 Represent the production for the years 2014 and 2015 as the matrices A and B. Find the difference in sales between the years 2014 and 2015. The company predicts that in the year 2020 the production at these factories will be double that of the year 2014.

Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix.

Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a “3 by 4 matrix”, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a 3×4 matrix does not have the same dimension as a 4×3 matrix. �=[14−203−1796205] �=[298−30165−2−478]

Types of Matrices Matrix addition Matrix addition explains the addition of two or more matrices. Unlike arithmetic addition of numbers, matrix addition will follow different rules. The order of matrices should be the same, before adding them. Before going into the addition of the matrix, let us have a brief idea of what are matrices. In mathematics, a matrix is a rectangular array of numbers, expressions or symbols, arranged in rows and columns. Horizontal Rows are denoted by “m” whereas the Vertical Columns are denoted by “n.” Thus a matrix (m x n) has m and n numbers of rows and columns respectively. We also know about different types of matrices such as square matrix, row matrix, null matrix, diagonal matrix, scalar matrix, identity matrix, diagonal matrix, triangular matrix, etc. Now, let us now focus on how to perform the basic operation on matrices such as matrix addition and subtraction with examples.

What is Matrix Addition? Addition of matrix is the basic operation performed, to add two or more matrices. Matrix addition is possible only if the order of the given matrices are the same. By order we mean, the number of rows and columns are the same for the matrices. Hence, we can add the corresponding elements of the matrices. But if the order is different then matrix addition is not possible. Suppose A = [aij]mxn and B = [bij]mxn are two matrices of order m x n, then the addition of A and B is given by;

A + B = [aij]mxn + [bij]mxn = [aij + bij]mxn

By recalling the small concept of addition of algebraic expressions, we know that while the addition of algebraic expressions can only be done with the corresponding like terms, similarly the addition of two matrices can be done by addition of corresponding terms in the matrix.

There are basically two criteria that define the addition of a matrix. They are as follows:

1. Consider two matrices A & B. These matrices can be added if (if and only if) the order of the matrices are equal, i.e. the two matrices have the same number of rows and columns. For example, say matrix A is of the order 3 × 4, then the matrix B can be added to matrix A if the order of B is also 3 × 4.

2. The addition of matrices is not defined for matrices of different sizes.

Properties of Matrix Addition The basic properties of matrix addition are similar to the addition of real numbers. Go through the properties given below:

Assume that, A, B and C be three m x n matrices, The following properties hold true for the matrix addition operation.

Commutative Property: If A and B are two matrices of the same order, say m x n, then the addition of the two matrices is commutative, i.e., A + B = B + A

Associative Property:: If A, B and C are three matrices of the same order, say m x n, then the addition of the three matrices is associative, i.e., A + (B + C) = (A + B) + C

Additive identity: For any m x n matrix, there is an identity element. Thus, if A is m x n order matrix, then the additive identity of A will be zero matrix of same order, such that, A + O = A ( where O is an additive identity)

Additive inverse: If A is any matrix of order m x n, then the additive inverse of A will be B (= -A) of same order, such that, A + B = O. Hence, the sum of matrix and its additive inverse results in a zero matrix

Subtraction of Matrices Subtraction of Matrices: The difference of two matrices is possible only when the order of the two matrices is the same. Similarly, addition of two matrices is done only when the order of the two matrices is the same. Thus, the resulting matrix will be of the same order.

The arithmetic operations on matrices are possible based on their order. Like the addition and subtraction of matrices requires the same order, but for multiplication of matrices, we need to check if the number of columns of one matrix is equal to the number of rows of the second matrix. The order of matrix multiplication is the order of the resulting matrix.

Definition of Subtraction of Matrices Mathematically, if there are two matrices, say A = [aij] and B = [bij] of the same order, say m × n, then the subtraction of A and B, i.e., A – B is defined as:

Matrix D = [dij]

A – B = aij – bij

Thus,

(dij = aij – bij, (i = 1,2,3,… and j= 1,2,3…)

D = A – B = aij – bij

Matrix Multiplication Definition Matrix Multiplication is a binary operation performed on two matrices to get a new matrix called the product matrix. Suppose we take two matrices A and B such that the number of columns in the first matrix is equal to the number of rows in the second matrix then we can multiply these two matrices to get a new matrix of the same order that is called the multiplication of the two matrices A and B. Thus, it is clear that not any two matrices can be multiplied and we can multiply only those matrices that follow a specific condition.

Matrix multiplication is the mathematical operation that is performed on two matrices which when multiplied gives a singular matrix. As we can perform multiplication on any two numbers we can not perform multiplication on any two matrices. We have to follow specific rules to perform matrix multiplication and the matrices obeying a certain relation are the matrices that can be multiplied.

If A and B are the two matrices, then the product of the two matrices A and B are denoted by:

X = AB

Hence, the product of two matrices is the dot prod