# Matrices, Objectives, Matrices and Their Representations, Matrices

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In the middle of the 19

^{th}Century, Arthur Cayley (1821-1895), an English mathematician created a new discipline of mathematics, called matrices. He used matrices to represent simultaneous system of equations.As of now, theory of matrices has come to stay as an important area of mathematics. The matrices are used in game theory, allocation of expenses, budgeting for by-products etc. Economists use them in social accounting, input-output tables and in the study of inter-industry economics.

Matrices are extensively used in solving the simultaneous system of equations. Linear programming has its base in matrix algebra. Matrices have found applications not only in mathematics, but in other subjects like Physics, Chemistry, Engineering, Linear Programming etc.

## Objectives

After studying this lesson, you will be able to:

Define a matrix, order of a matrix and cite examples thereof

Define and cite examples of various types of matrices-square, rectangular, unit, zero, diagonal, row, column matrix

State the conditions for equality of two matrices

Define transpose of a matrix

Define symmetric and skew symmetric matrices and cite examples

Find the sum and the difference of two matrices of the same order

Multiply a matrix by a scalar

State the condition for multiplication of two matrices

Multiply two matrices whenever possible

Use elementary transformations

Find inverse using elementary transformations

## Matrices and Their Representations

Suppose we wish to express that Anil has pencils. We may express it as or with the understanding that the number inside denotes the number of pencils that Anil has.

Next suppose that we want to express that Anil has books and pencils. We may express it as with the understanding that the first entry inside denotes the number of books; while the second entry, the number of pencils, possessed by Anil.

Let us now consider, the case of two friends Shyam and Irfan. Shyam has books, notebooks and pens; and Irfan has books, notebooks and pens.

A convenient way of representing this information is in the tabular form as follows:

Books | Notebooks | Pens | |

Shyam | |||

Irfan |

We can also briefly write this as follows:

First Column Second Column Third Column

This representation gives the following information:

(1) The entries in the first and second rows represent the number of objects (Books, Notebooks and Pens) possessed by Shyam and Irfan, respectively

(2) The entries in the first, second and third columns represent the number of books, the number of notebooks and the number of pens, respectively.

Thus, the entry in the first row and third column represents the number of pens possessed by Shyam. Each entry in the above display can be interpreted similarly.

The above information can also be represented as

Shyam | Irfan | |

Books | ||

Notebooks | ||

Pens |

Which can be expressed in three rows and two columns as given below:

The arrangement is called a matrix. Usually, we denote a matrix by a capital letter of

English alphabets, i.e. , etc. Thus, to represent the above information in the form of a matrix, we write

Order of a Matrix Observe the following matrices (arrangement of numbers):

(a) (b) (c)

In matrix (a), there are two rows and two columns, this is called a by matrix or a matrix of order . This is written as matrix. In matrix (b), there are three rows and two columns. It is a by matrix or a matrix of order . It is written as matrix. The matrix (c) is a matrix of order .

Use of two suffixes and helps in locating any particular element of a matrix. In the above matrix, the element belongs to the row and column.

A matrix of order can also be written as

and

Example:

Write the order of each of the following matrices:

(a)

(b)

Solution:

The order of the matrix

(a) is

(b) is