Sequence & Series

First Year Science

Mathematics

Unit 5: Linear Algebra

5.1 Sequence & Series

Introduction:Definition: A real sequence (sequence) may be defined as a function. It is said to be finite or infinite according as the number of elements in it is finite or infinite.

A finite sequence is written as or as or simply and an infinite sequence as . In both cases, the term is known as the general term or term of the sequence.Definition: Let be a given sequence, then the sum

is called a finite series and is written as . Similarly, the corresponding infinite series is defined.Definition: Special sequences or series, in which the terms occur in a definite order, are known as progressions.Mainly, there are three types of progression: i) Arithmetic progression (A.P.), ii) Geometric progression (G.P.) and iii) Harmonic progression (H.P.) defined as follows:

is an A.P.

is constant for all . Here, the value of is

called the common difference. is a G.P.

EMBED Equation.DSMT4 is constant for all . Here, is called the common ratio.

is an H.P.

EMBED Equation.DSMT4 for all .

Examples: i) is an A.P with common difference (c.d.), . ii) is a G.P. with common ratio (c.r), .

iii) is an H.P.

General Terms:(i) The term of an A.P. with first term and the common difference is given by the formula: . In the case of a finite A.P. , the last term is given by .

(ii) The term of a G.P. with first term and the common ratio is given by the formula: .

Examples: i) The 50th term of the sequence 2, 5, 8, 11, is

ii) The 10th term of the sequence 1, 2, 4, 8, is .

Sum of n Terms:

(i) The sum of the first n tems of an A.P. with first term and common difference is given by the formula: .

(ii) The sum of the first n terms of a G.P. with first term and common ratio is given by the formula: .Example1 Find the sum of the first 35 terms of an A.P., whose second term is 2 and seventh term is 22.

Solution: We know that .

i.e (i) (ii).Solving these two equations, we get: .

Now, using the formula , we get

.

The required sum is 2310.

Example2: If you save Rs. 1 today, Rs. 2 tomorrow, Rs. 4 the day after tomorrow and so on, what will be your total savings in two weeks?

Solution: Total savings in two weeks

= Rs.(to 14 terms)

= Rs. [] = Rs.

= Rs.16383.Example3: If 7 times the 7th term of an A.P. is equal to 11 times the 11th term, show that

the 18th term of the A.P. is equal to 0.

Solution: Let be the first term and be the comon difference of the A.P., then

EMBED Equation.DSMT4 .By question,

EMBED Equation.DSMT4

EMBED Equation.DSMT4 .Now, . That is .

Example 4: The sum of Rs. 6240 is paid off in 30 installments such that each installment is Rs.10 more than the preceding installment. Calculate the value of the first installment.Solution: Let be the first installment. Clearly, the installments are in A.P. with common difference

We know that .

By question,

Now,

.

Hence the required value is Rs.63.Example5: Sum to n terms: 4 + 44 + 444 +

Solution: to terms

= to terms)

= , to terms = , to terms = , using the formula for the sum of a G.P.

= .

Required sum is .

Exercise 5(A)1. Write the first four terms of each of the sequences, whose general term is

(i) (ii) (iii) .

2. Find the general term in each of the following sequences:

(i) (ii) (iii)

(iv) (v)

3. If the 9th term of an A.P. is 99 and 99th term is 9, find the 108th term of the A.P.

4. The 4th, 7th , 10th terms of a G.P.are respectively, prove that

5. A man repays a loan of Rs. 3250by paying Rs. 20in the first month and then increases

the payment by Rs. 15 every month. How long will take to clear his loan?6. A club consists of members, whose ages in years are in A.P., the common difference

being 3 months. If the youngest member of the clubis just 7 years old and the sum of

the ages of all the members is 250 years, find the number of members in the club.

7. Find the sum of the series to 12 terms.

8. Sum to n terms of the series: .

Properties of A.P., G.P., and H.P.[A].If is an A.P. and is a non-zero constant, then each of the following

sequences is also an A.P. i) ii) .

iii) iv) .[B].If is a G.P. and is a non-zero constant, then each of the following sequences is also a G.P.

i) ii)

iii) iv) .[C]. If is an H.P. and is a non-zero constant, then each of the following

sequences is also an H.P.

i) ii)

Example: If are in A.P. and , then prove that

are also in A.P.Solution: are in A.P

are in A.P.

are in A.P.

EMBED Equation.DSMT4 are in A.P. Sum of Infinite G.P.Formula: The sum of an infinite G.P. with first term and common ratio is given by , provided that .Proof: The sum of the first terms of the G.P. is .

Now, taking limits on both sides as , we get:

EMBED Equation.DSMT4 = = =.Hence, the formula for the sum of an infinite G.P. is: , when

Note that the sum does not exist when .

Example: Find the following sums whenever exist:

(i) to (ii) to

EMBED Equation.DSMT4 Solution: (i) The series is a G.P. with first term 1 and common ratio 2. Since , therefore the infinite sum does not exist.

(ii) In this case, . The infinite sum exists and it is given by .

Hence the required sum is .

Arithmetic, Geometric & Harmonic MeansDefinition: If is an A.P., then the middle term is called the arithmetic mean between the numbers . In short, we denote this mean by A.M. The geometric and harmonic means are defined in similar way.

The formulae for these means are as follows:

(i) (ii) (iii) .Remark: By using the formulae, it can be easily proved that

In general, the arithmetic means between two numbers are given by

, where the common difference =.

The formulae for the geometric means between are:

. Here the common ratio is given by .

Example: Insert 6 geometric means between

Solution: Here,

The common ratio can be found as follows:

EMBED Equation.DSMT4

EMBED Equation.DSMT4 = 3.That is, . Now, the required geometric means are found as shown below.

and so on.The required means will be

Sum of Natural NumbersThe following three formulae are very useful:

(i) The sum of the first n natural numbers, . That is, .

(ii) The sum of the squares of the first n natural numbers, . That is, .(iii) The sum of the cubes of the first n natural numbers,

. That is, .

Note that these formulae can be established by the method of induction.

Example: Sum to n terms of the series

Solution: The term of the given series is

Now, the required sum is

=

=

= .

.Exercise 5(B):1. The product of three numbers in G.P. is equal to 216. If 2, 8, 6 are added to those numbers respectively, the resulting numbers are in A.P. Find the numbers.

2. The A.M between two numbers, whose sum is 100, is to the G.M as 5:4, find the numbers.3. Insert four arithmetic means between , then verify that their sum is 7.

4. Insert 3 geometric means between

5. Sum to n terms of the following series:

(i) (ii)

6. Find the sum of the following infinite series, if they exist:

(i) (ii) 6 + (0.6) + (0.06) + (0.006) + (iii) (iv)

7. The sum of the first two terms of a G.P. is 4 and the sum of their squares

(i.e squares of the terms) is 8. Find the G.P.

8. A ball is dropped from a height of 48 feet and it rebounds two third of the height

it falls. If it continues to fall and rebound infinitely, find the total distance that the ball can travel before coming to rest. 9. A person is entitled to receive an annual payment which for each year is less by

one-tenth of what it was for the year before. If the first payment is Rs. 800, show

that he/she can not receive more than Rs. 8000 however long he/she may live.

5.2 Permutations & Combinations

Basic Principle of Counting

Suppose that there are a number of operations to be performed. If the1st operation can be done in ways, 2nd in ways, 3rd in ways and so on, then all these operations together can be performed in ways.Example: In how many ways can a Chairman and a secretary be elected from a committee of 9 members, who are equally eligible for the two posts?Solution: Since any one of the 9 members may be elected as a chairman, the Chairman can be elected in 9 ways. After that out of the remaining 8 members; the secretary can be elected in 8 ways. So, by the basic pinciple of counting, the required number of ways is equal to.

Permutations & CombinationsDef 1: An arrangement that can be made out of a given number of objects by taking some or all of them at a time is called a permutation.For example; the permutations of the three letters A,B,C taken two at a time are: AB, BA, BC, CB, AC, CA. So, there are 6 permutations. Def 2: A collection that can be made out of a given number of objects by taking some or all of them at a time is called a combination.For example; the number of combinations of the letters A, B, C taken two at a time is just 3. The combinations are: AB, BC, CA . Here, BA & AB are the same combinations. Similarly, CB & BC are the same, and CA & AC are also the same.

Note that in permutations, the order of the objects is taken into account, but in combinations the order is not considered.(i) The number of permutations of different things taken at a time is denoted by . (ii) The number of combinations ofdifferent things taken at a time is denoted as .

(iii) The continued product of natural numbers is denoted by the symbol

(Read factorial) or n! (Read factorial). For e.g .

Formulae(i) The total number of permutations of different things taken at a time is given by

the formula: , where .

(ii) The total number of combinations of of different things taken at a time is given by

the formula:

, where .

Remarks:

- In particular, the number of permutations of different things taken all at a time is given by Also,

So, and .- Similarly, we can have the following results:

and . Properties of

By using the definitions, the following properties can be established:

(i) . These two combinations are known as complementary

combinations. Note that if , then either or.

(ii) .

Example: If , determine the value of and hence find

Solution: Since , therefore either (impossible) or i.e Now, .

Special casesGiven below are some of the special cases in permutations and cominations: (i) Permutations of things not all different:

The number of permutations of things, among which things are alike of one kind, are alike of second kind, are alike of third kind, and the rest different is equal to .Example: How many arrangements can be made out of the letters of the word MATHEMATICS, taken all at a time?

Solution: There are altogether 11 letters in the give word, out of which there are 2 Ms, 2 As, and 2 Ts, others are all different. So, the number of ways in which the 11 letters can be arranged is equal to.

(ii) Permutations of things with repetition:

The number of permutations of different things taken at a time, when repetition of things is allowed is equal to .

Suppose, for example, that we have to post 5 lettersin in 10 letter boxes.

Obviously, each letter can be posted in 10 differst ways. So, the number of ways to post all the 5 letters wll is

(iii) Circular Permutations:

In a circular permutation, one of the given things is to be kept fixed and the remaining things are kept relative to it. So, the number of ways, in which different things can be arranged in a circle taking all at a time, is

Example: In how many ways can 8 students form a ring?

Solution: Since we are concerned with the relative positions of the students , therefore one student is to be kept fixed. Now, the remaining 7 students can be arranged in 7! ways.

Hence, the required number of ways is 7! = 5040.

Remark: There is a little difference between the case of people around a table and the case of beads in a necklace. What is that? Beads are the same whether you look at them from one side or the other. Because of this, there may be two arrangements which initially look different, but become the same if one of them is turned around. It can not be done with people.So, in the case of beads, the number of circular arrangements of beads is .Example: In how many ways can 6 different beads be strung into a necklace?

Solution: In general, 6 different things can be arranged in a circle in ways. However, in the case of the circular arrangements of beads to form a necklace, the clockwise and counter-clockwise orders are not distinguished.So, the required number of ways is .

(iv) Conditional permutations & combinations:a) The number of permutations of different things taken at a time, when particular things always occur is equal to

b) The number of permutations of different things taken at a time, when

particular things never occur is equal to

c) The number of combinations of different things taken at a time, when

particular things always occur is equal to

d) The number of combinations of different things taken at a time, when

particular things never occur is equal to

Example1: How many three - letter words can be formed using the letters of the word FAILURE such that (a) L is always present (b) U is never present.Solution: (a) The required number is .

(b) The number of desired words is .

Example 2: From a committee of 7 men and 5 women, in how many ways can a

subcommittee of 4 members be chosen so as (a) to contain one perticular

person (b) not to contain two perticular persons in the committee?

Solution: (a) After keeping aside the particular person, we chooe 3 more members from the remaining 11. It can be done in ways. (b) Leaving aside the two particular persons, we are to choose 4 persons

from the remaining 10, which can be done in ways.

Exercise 5(C):1. There are 8 doors in a college hall.In how many ways can a student enter the hall through one door and come out in 7 different ways? [ 56 ] Suppose there are 12 persons in a party. If each of them shakes hands with each other, how many handshakes happen in the party? [ 66 ]2. How many odd numbers of two different digits can be formed from the integers

1, 2, 3, 4? [6]3. From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition, at least including 4 boys and 4 girls. The two girls who won the prizes last year should be included. In how many ways can the selection be made?

[108474]

4. In a restaurantthe menu lists 5 vegetables, 3 meats, two salads and 4 breads. In how many ways can a customer make a meal consisting of a unit of each item?

[120]

5. How many different numbers between 4000 and 5000 can be formed with the digits 2, 3, 4, 5, 6 and 7? [60]6. If are in A.P., find the value(s) of [7 or 14]7. A student has 6 friends. In how many ways may he/she invite one or more of them to a dinner? [63]8. In how many ways can a committee of 3 ladies and 4 gentlemen be appointed from a meeting consisting of 18 ladies and 14 gentlemen? [816816]9. A candidate is required to attempt 6 out of 10 questions divided into two groups of 5 questions each. If he/she should not attempt more than 4 questions from each group, find the number of different ways in which he/she can choose the 6 questions. [200]10. There are 3 sections in a question paper, each containing 5 questions. A candidate has to solve any 5 questions choosing at least one question from each section. In how many ways can he/she make the choice? [2250] 11. There are 5 gentlemen and 4 ladies to dine at a round table. In how many ways

can they seat themselves so that no two ladiesare together? [2880]

12. In how many ways can 6 boys and 5 girls be arranged for a group photograph, if the girls are to sit on chairs in a row and the boys are to stand behind them? [86400]

13. Find the number of arrangements that can be made out of the letters of the following words: (i)ACCOUNTANT (ii) MATHEMATICS (iii) MISSISSIPPI

[ (i) 226800 (ii) 4989600 (iii) 34650 ]

14. How many 4-letter code symbols can be formed with the letters A, B and C, when

the repetition is allowed? [81]

15. If the number of 4 permutations of n things, in which one particular thing does not

occur, is equal to the number in which it does occur, find n. [8]

16. In how many ways can 7 persons be selected from 16 so that (i) 4 particular

persons will not be there (ii) 4 particular persons will always be there? [ (i) 792 (ii) 220 ]

5.3 Binomial Theorem

Introduction:

By actual multiplication, we have:

Here, we can notice that the coefficients of terms have a certain pattern.

An expression of type or is called a binomial. The formula, first discovered by Newton, for the power of a binomial is known as Newtons formula or Binomial therem as stated below: Statement of the Theorem:

Let Then the following formula holds true:

The following observations are straightforward:

(i) The number of terms in the expansion is equal to .

(ii) The sum of the powers of in each term is equal to .(iii)The coeficients of terms equidistant from the beginning and the end are equal.

Example: Use Binomial Theorem to expand .Solution: Using Binomial Theorem, we get:

=

= =

General Term:In the expansion of by Binomial theorem, the term is given by the formula: . This term is caled the general term of the expansion as we can obtain all the terms from it by putting .

Example: Find the term independent of in the expansion of . Solution: The general term is given by = . This term is to be independent of

EMBED Equation.DSMT4

The term, i.e the term is independent of and it is equal to Hence, the required term is 84. Remark: In various practical problems, especially in approximations, we need the use of Binomial expansion for where is a negative integer or a fraction (positive or negative). In particular, we consider the expansion formula for:

, where

Example: Compute correct to three decimal places.

Solution:

EMBED Equation.DSMT4

, which is correct to three places after decimal. Exercise 5(D): 1. How many terms are there in the expansion of 2. Use Binomial theorem to expand the followings:

(i) (ii) . 3. Find the coefficient of in the expansion of .

4. Find the general term in the expansion of .

5. Find the term independent of in the expansion of . 6. If in the expansion of the coefficients of are equal,

find the value(s) of .

7. The coefficients of the second, third and fourth terms in the expansion of

are in A.P., find the value of .

8. Using Binomial theorem, find the value of (i) (ii) (iii) correct to 4 decimal places.

9. Compute the followings to 3 significant figures:

(i) (ii) . 10. Find the first three terms in the expansion of

5.4 MatricesThe theory of matrices has become one of the most powerful tools in mathematics. It has wide applications in various disciplines such as physical sciences, social sciences, engineering, statistics, business and management.

Basic Definitions: Def.1: A matrix (plural form - matrices) is an arrangement of numbers in a

rectangular form consisting of one or more rows and columns. Each number in the arrangement is called an entry or element of the matrix. A matrix is usually denoted by a capital letter and its elements are enclosed within square brackets [ ] or round brackets ( ) or double verticall bars. If denotes the element in the row and column of a matrix , then the matrix is written in the following form:

EMBED Equation.DSMT4 .

Def.2: If a matrix has m rows and n columns, we call it a matrix of order m by n. the order of a matrix is also known as the size or dimension of the matrix. For example; the matrices

are of the orders respectively.

Def.3: Two matrices are said to be equal to each other if and only if they are of the

same order and have the same corresponding elements. BasicTypes of Matrices:1. Row matrix a matrix having a single row. For e.g; .

2. Column matrix a matrix having a single column. For e.g;

EMBED Equation.DSMT4 .

3. Null matrix a matrix having all elements zero. For e.g;

A null matrix is also known as a zero matrix, and it is usually denoted by O.4. Square matrix a matrix having equal number of rows and columns. For e.g; the matrix is a square matrix of size .

5. Diagonal matrix a square matrix, all of whose elements except those in the leading diagonal, are zero. For e.g; .

6. Scalar matrix a diagonal matrix having all the diagonal elements equal to each other. For e.g; .

7. Unit matrix a diagonal matrix having all the diagonal elements equal to 1.

For e.g; , , A unit matrix is also known as an identity

matrix and is denoted by the capital letter I.8. Triangular matrix a square matrix, in which all the elements below (or above) the leading diagonal are zero.

For e.g; & are upper triangular and lower triangular matrices respectively.9. Symmetric matrix a square matrix such that

For e.g; .

10. Skew-symmetric matrix - a square matrix such that

For e.g; . Note that the elements in the leading diagonal of a skew-symmetric matrix are always zero.

Basic Operations on Matrices: We can obtain new matrices from the given ones by using the following

operations.1. Addition & subtraction If are matrices, then their sum is defined as the new matrix: , where . The order of this sum is again . Similarly, is defined. Note that the sum or difference of two matrices is defined only when the matrices

have the same size.

Example: If, then

are defined, whereas & are not defined.2. Scalar multiplication If is a matrix and is a scalar, then the scalar multiple of by , denoted by, is the matrix defined by . So, to multiply a given matrix by a constant means to multiply each element

of by. Example: Let and . Then, we get:

.

3. Multiplication If and are two matrices of orders and respectively, then the product is the new matrix of order defined by the formula: , where . In short, we write: . Example: Let. Then, we get:

EMBED Equation.DSMT4 .4. Transposition The transpose of an matrix is defined as the matrix, where. It is also denoted as.

Example: If then.

Remarks: (i) is symmetric

EMBED Equation.DSMT4 = (ii) is skew-symmetric

EMBED Equation.DSMT4 = -.

Properties of Matrix Operations: Let be given matrices, then the basic properties of matrix addition, scalar multiplication, matrix multiplication and matrix trasposition are stated below without proof. These properties can be easily verified in examples.

1. Properties of matrix addition and scalar multiplication:

i) [Commutativity]

ii) [Associativity]

iii), where is the corresponding null matrix.

iv) , where is a scalar [Distributivity]2. Properties of matrix multiplication:

i) , in general

ii) [Associativity]

iii), where is the corresponding identity matrix.

iv) [Distributivity]

3. Properties of matrix transposition:

i)

ii)

iii)

iv) .

Example1: Given the matrices , verify that .

Solution: We have, by multiplication: .

. So,.

Example 2: Given matrix is

EMBED Equation.DSMT4 . Verify that

Solution: Here,

EMBED Equation.DSMT4

. Hence, it is verified.

Example 3: Suppose that a company puts a problem to you: Determine which of the three methods of production it should use in producing three goods. The amount of each good produced by each method is shown below in matrix form: A B C

.

The row matrix represents the profit per unit for the goods respectively. Use matrix multiplication to decide which method maximizes the total profit.

Solution: Let

EMBED Equation.DSMT4 & . Then, .

Now, the total profits from the three methods are given by

=.

So, the profits obtained from the methods are Rs. 516, Rs. 765 & Rs. 564 respectively. Consequently, the second method maximizes the profit. Exercise 5(E)1. If , find the values of

2. Construct a matrix, whose elements are given by the formula:.

3. If and ,

find , if exist.

4. If is a square matrix, prove that are symmetric matrices.5. A manufacturer produces three products which he sells in two markets annual sale volumes are indicated as follows:

P Q R

EMBED Equation.DSMT4 . (i) If the sale prices of the products are Rs. 2

respectively, find the total revenue in each market using matrices.

(ii) If the costs of the above three products are Rs respectively,

find the gross profit.6. A new company is carrying outanalysis on three alternate methods of producing three products. Due to the variation in cost of productionthe profits due to the three methods vary. If profit maximization is the objective of the compan, make a suggestion from the following information, which method should be adopted by the company. The amount of goods (in 000 unit) produced by each method is as shown below:

.

The profit per unit (Rs. 000) from each good is estimated to be [100, 145, 80].5.5 Determinants:The determinant of a square matrix is denoted by the symbol or . We can form determinants of matrices. Such determinants are called determinants.

Definition: i) If is a matrix, then its determinant is equal to the number itself.

ii) If is a matrix, then the determinant is given by

iii) If is a matrix, then its determinant is given by

EMBED Equation.DSMT4 = .

Note that the determinant is expanded along the first row. Similarly, the determinant can be expanded along any other row or column carrying (+) or (-) sign according to the place occupied by the element in the following scheme: Example:

(i) (ii) =

(iii) , expanding along the first row

=.

Sarrus rule:

To find the value of a determinant, the following rule, called the Sarrus rule may also be useful.(i) Consider the determinant: .(ii) Write the three columns and then repeat the first two to have the fourth and fifth columns respectively as follows (vertical bars are to be omitted):

(iii) Write the product of the elements of each leading diagonal with positive sign. Also write the product of the elements of each secondary diagonal with negative sign.(iv) The sum of the products obtained in step (iii) gives the value of the determinant.

Example: Use Sarrus rule to find the value of the following determinant:

Solution: The elements of the given determinant are arranged as follows:

Now, according to Sarrus rule, the value of the determinat is

=

=

= 14. Minors & CofactorsDefinition: Let D = be a determinant. Then the determinant obtained by deleting the row and column, in which an element lies, is called the minor of the element and is denoted by .

Definition: If is the minor of an element of a determinant, then the cofactor of the element, denoted by is defined as

Thus, the minor of in D is = and the cofactor is .Remark: The value of the determinant D can be expressed in terms of the elements of any row or column of D.

Example: Write the cofactors of elements of second row of the determinant

and hence find the value of the determinant.

Solution: Here,

So, . Now, .

Properties of DeterminantsFollowing properties may be helpful to find the values of determinants:

P1: If any two rows or columns of a determinant are identical, then its value equals zero.

Example: [The first and second rows are identical.]P2: If all the elements in any row or column of a determinant are zero, then the value of the determinant is alo zero. Example: [The elements of third row are all 0.]

P3: The determinant of a unit matrix is equal to 1.

Example: .P4: The determinant of a diagonal matrix is equal to the product of the diagonal elements.

Example: .P5: The determinant of a square matrix equals the determinant of its transpose.

Example: .P6: If any two adjacent rows or columns of a determinant are interchanged, then the value of the determinant changes by sign.

Example: .P7: If each element of a row (or column) of a determinant is multiplied by a scalar k,

then the value of the determinant is also multiplied k.

Example: .P8: A determinant of the form can be expressed in the form of the sum of

two determinants:

P9: If a multiple of any row (or column) of a determinant is added to (or subtracted from)

any other row (or column), then the value of the determinant remains unchanged.

Example:

P10: The determinant of a product of two square matrices is equal to the product of their

determinants. That is, . Further Examples:(1) Evaluate the following determinants as indicated:

(i) (along the first row) (ii) (along the first column).Solution: (i) = 1

(ii) = 1.(2) Evaluate the following determinant without expanding:

Solution: ;

= ; taken common from

= = 0.(3) Prove that

Solution:

= expanding the determinant along

= ; taking commons =

=

(4) Solve the following equation:

.

Solution: Here, we have to find the value of

Now,

EMBED Equation.DSMT4

Exercise 5(F)1. Is it possible to define the determinants of the following matrices? Evaluate whenever possible. (i) (ii) (iii) (iv) .

2. Evaluate the following determinants:

(i) (ii) (iii) .3. Use Sarrus rule to find the values of

(i) (ii) (iii) .

4. Find the minor and cofactor of each element of the determinant.

5. Use the properties of determinants to show that each of the following determinants

is equal to zero:

(i) (ii) (iii) .6. Prove, without expanding, that

(i) = (ii)

(iii) (iv)

7. Prove the following results:

(i) (ii)

(iii) . 8. Solve the following equations:

(i) (ii) (iii)

Inverse Matrix

Definition1: A square matrix is said to be singular, if its determinant and non-singular, if .

Definition 2: The adjoint of a square matrix, denoted as Adj.A, is defined as the transpose of the matrix obtained by replacing each element of by its cofactor.

So, Adj. = .

Example: Let . Then, the cofactors of its elements are: So, the matrix of cofactors is

Definition 3: If are square matrices such that , where is the unit matrix of the same order, then is called the inverse (or reciprocal) of and is denoted by . Similarly, is said to be the inverse of. Thus, and .Formula: The inverse of a non-singular matrix is given by the formula:

Note that no inverse of exists, when

Example: Find the inverse of the matrix

Solution: Let . Then, the determinant of A is: .Since exists.

Now, the cofactors of the elements of are:

The matrix of cofactors is & hence .

Now, .

Exercise 5(G)1. Find the matrix of cofactors in each of the following matrices:

(i) (ii) .

2. Find the adjoint of each of the following marices:

(i) (ii) . 3. If , verify that

4. If , show that

5. Obtain the matrices , when .

6. Find the matrix, whose inverse is .

7. Find the inverse of the matrix and verify that where is the unit matrix of order 3.

8. Show that the following matrices are inverses of each other:

.

5.6 System of Linear Equations: Basic DefinitionsDef 1: An equation of the form , where at least one of the real numbers is not zero, is called a linear equation in . Similarly, an equation of the form is a linear equation in three variables

Def 2: A system of equations can have exactly one solution, no solution or an infinite number of solutions. A system with at least one solution is called a consistent system. If it has unique solution, it is said to be consistent and independent.Example: The system has exctly one solution. The unique solution is: . So, it is a consistent and independent system.

Def 3: A system of equations having no solution is called an inconsistent system.

Example: The system has no solution, because no values of satisfy both equations. Therefore, it is an inconsistent system.Def.4: A system of equations is said to be consistent and dependent, if it has infinitely many solutions. Example: The system is consistent, but dependent.

Methods of SolutionI. Row-equivalent matrix method: According to this mehod, to solve a system of linear equations in two variables, say, , we form the following matrix, called augmented matrix: Then, we use row operations to change this matrix into row equivalent matrices. We use the symbol to indicate that are row-equivalent matrices. Some of the elementary row operations are as follows:(i) Interchange of any two rows, for e.g, .

(ii) Multiplication of each element of a row by a non-zero number,

for e.g, .

(iii) Multiple of a row added to (or subtracted from) any other row,

for e.g, .

In this way, row operations are performed until we get the following special form:

. Then, the solution of the system will be

Note: A system of three linear equations with three variables can be solved similarly.

Example 1: Find the solution, if any, of the system .

Solution: The augmented matrix is

.

Thus the required solution is:

Example 2: Solve the following system of equations:

.

Solution: Starting with the corresponding augmented matrix and using elementary row-operations, we get the following chain. ,

The required solution is

II. Inverse Matrix Method:

Consider the following system of linear equations:

This system can also be written in the form .

So, we have:, where .

Now,

For a system of three linear equations:

, we have:

So, the relation can be used to determine the values of

that satisfy the system.

Example1: Solve the system.

Solution: The given system can be written in the form , where The solution is given by

Lets first find.

Here, &

.

Now,

EMBED Equation.DSMT4

.Hence the required solution is:

Example2: Find the solution, if any, of the following system: .Solution: The given system can be written as

.

So, we have: , where

. Lets find first. The cofactors of the elements of are:

.

The matrix of cofactors is .

Again,

.

Now,

.Hence the required solution is:

III. Determinant Method (Cramers Rule):Let us consider the system of equations: .

Multiplying the first equation by , second by and then subtracting the second from the first, we get: .

Similarly, we obtain , provided that

Alternatively, we can write the above mentioned formulae as shown below:

If

then the values of are given by the formulae:

.In the same way, for a system of three linear equations: ,

the values of are given by the formulae: , where

.It should be noted here that .

Example 1: Use Cramers rule to solve the equations:

Solution: we have:

Now, by Cramers rule, we get:

Example 2: Use the determinant method to solve the following system of linear equations: .

Solution: First of all, we calculate the following determinants:

& Now, by Cramers rule, we have:

.

Hence, the required solution is

Exercise 5(H)1. Solve the following systems of equations by Row-equivalent matrix method:

(i)

EMBED Equation.DSMT4 (ii)

(iii) (iv) .2. Solve the following systems of equations by using inverse matrices:

(i) (ii)

(iii) (iv) .

3. Use Cramers rule to solve the folowing systems of equations:

(i) (ii)

(iii) (iii) .PAGE 1 | Page

Prepared By: Sir Muhammad Asad Ali

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