CU-BCA-SEM-I-Mathematics (1)

BACHELOR OF COMPUTER APPLICATIONS SEMESTER - I MATHEMATICS

First Published in 2021 All rights reserved. No Part of this book may be reproduced or transmitted, in any form or by any means, without permission in writing from Chandigarh University. Any person who does any unauthorized act in relation to this book may be liable to criminal prosecution and civil claims for damages. This book is meant for educational and learning purpose. The authors of the book has/have taken all reasonable care to ensure that the contents of the book do not violate any existing copyright or other intellectual property rights of any person in any manner whatsoever. In the event the Authors has/ have been unable to track any source and if any copyright has been inadvertently infringed, please notify the publisher in writing for corrective action. 2 CU IDOL SELF LEARNING MATERIAL (SLM)

CONTENT 4 31 Unit - 1: Matrix 1 47 Unit - 2: Matrix 2 68 Unit - 3: Matrix3 88 Unit - 4: Matrix4 111 Unit - 5: Matrix 5 124 Unit - 6: Linear Programming 1 134 Unit - 7: Linear Programming 2 150 Unit - 8: Different Types Of Linear Programming (L.P.) Problems 1 176 Unit - 9: Different Types Of Linear Programming (L.P.) Problems 2 193 Unit - 10: Different Types Of Linear Programming (L.P.) Problems 3 205 Unit - 11: Combinations 1 217 Unit - 12: Combinations 2 226 Unit - 13: Propositional Logic 1 241 Unit - 14: Propositional Logic 2 Unit - 15: Propositional Logic 3 3 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 1: MATRIX 1 STRUCTURE 1.0 Learning Objectives 1.1 Introduction 1.2 Definition of a matrices 1.3 Types of Matrix 1.4 Operation of Matrices 1.4.1. Addition and Subtraction of Matrices 1.4.2. Scalar multiplication. 1.5 Summary 1.6 Keywords 1.7 Learning Activity 1.8 Unit End Questions 1.9 References 1.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Describe the required conditions for matrix addition and subtraction ● Understand the properties of matrix addition and subtraction ● Discuss the properties of the identity matrix ● Explain how to use row operations and why they produce equivalent matrices ● Solve problems involving scalar multiplication. ● Understand the properties of matrix scalar multiplication. 1.1 INTRODUCTION The solution of linear equations and their applications are addressed in this chapter. The term \"matrix\" was coined in 1850 by author James Joseph Sylvester, who defined it as an item that generates a number of minors (determinants), that is, determinants of smaller matrices generated from the original one by eliminating columns and rows. After that another Mathematician Culli's was use to modern bracket notation for matrices and also he introduced simultaneously to the first significant use of the notation A=[aij] to represent a matrix where aij refers to the element found in the ith row and the jth column. Matrices can be used to 4 CU IDOL SELF LEARNING MATERIAL (SLM)

write compactly and easy to work for given multiple linear equations and referred to as a system of linear equations. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A matrix is defined by some rectangular array of m x n real or complex numbers in the form of m horizontal lines (rows) and n vertical lines (columns) and it is a matrix of order m by n, also written as m x n matrix. Generally, we can used an array is enclosed by [ ] or ( ). An m x n matrix is usually written as:  a11 a12 a13 ..... a1n     a21 a22 a23 ..... a2n   a31 a32 a33 ..... a3n           A= am1 am2 am3 ..... amn  In brief, the above matrix is represented by A = [aij]mxn. The number a11, a12, ….. etc., are known as the elements of the matrix A, where aij belongs to the ith row and jth column and is called the (i, j)th element of the matrix A = [aij]. 1.2 DEFINITION OF MATRIX Definition: An ordered rectangular array of numbers or functions is called matrix where thehe numbers or functions are called the elements of the matrix. For example, matrix A has two rows and three columns. Order of a Matrix: Definition: If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn number of elements. In general,am×n matrix has the following rectangular array: 5 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: The order or dimensions of matrix is 4 X 3. 1.3 TYPES OF MATRICES Column Matrix: If a matrix Ais called a column matrix then it has only one column.  2   1 e.g.  5  In general, A = [aij]m×1 is a column matrix of order m × 1. Row Matrix: If a matrix A is called a row matrix, then it has only one row. e.g., 2 5  6 In general, A = [aij]1×n is a row matrix of order 1 x n Square Matrix: If a matrix A is called a square matrix, then it has equal number of rows and columns. 5 6 3 2 8 7 e.g. 1 4 6 In general, A = [aij]m x m is a square matrix of order m. Note: If A = [aij] is a square matrix of order n, then elements a11, a22, a33,…, ann is said to constitute the diagonal of the matrix A. Diagonal Matrix: A square matrix whose, all the elements except the diagonal elements are zeroes, is called a diagonal matrix. 6 CU IDOL SELF LEARNING MATERIAL (SLM)

 2 0 0 0 3 0 e.g. 0 0 6 In general, A = [aij]m×m is a diagonal matrix, if aij = 0, when i ≠ j. Scalar Matrix: A diagonal matrix is called a scalar matrix, whose all diagonal elements are same (non-zero). 3 0 0 0 3 0 e.g. 0 0 3 In general, A = [aij]n×n is a scalar matrix, if aij = 0, when i ≠ j, aij = k (constant), when i = j. Note: A scalar matrix is a diagonal matrix, but a diagonal matrix may or may not be a scalar matrix. Unit or Identity Matrix: A diagonal matrix in which all diagonal elements are ‘1’ and all non-diagonal elements are zero, is called an identity matrix and it is denoted by I. 1 0 0 0 1 0 e.g., 0 0 1 In general, A = [aij]n×n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j. Zero or Null Matrix: A matrix is said to be a zero or null matrix, if its all elements are zero. 0 0 0 0 0 0 e.g., 0 0 0 Equality of Matrices: Two matrices A and B are said to be equal, if (i) order of A and B are same. (ii) corresponding elements of A and B are same i.e., aij = bij, ∀ i and j. 7 CU IDOL SELF LEARNING MATERIAL (SLM)

1 2 0 1 2 0  2  6 9 3  6 0 3  6 0  0 8 5 e.g. 0 5 9 and 0 5 9 are equal matrices, but   2 3 4  4  6 9  0 8 6 and   2 3 4 are not equal matrices. Symmetric and Skew-Symmetric Matrices 1. A square matrix A = [aij]n x n, is said to be symmetric, if AT = A. i.e., aij = aji , for all i and j. 2. A square matrix A is said to be skew-symmetric matrices, if A = AT i.e., aij = - ajifor all i and j. Properties of Symmetric and Skew-Symmetric Matrices 1. Elements of principal diagonals of a skew-symmetric matrix are all zero. i.e., aii = -aii or aii = 0, for all values of i. 2. If A is a square matrix, then (a) A + ATis symmetric. (b) A -AT is skew-symmetric matrix. 3. If A and B are two symmetric (or skew-symmetric) matrices of same order, then A + B is also symmetric (or skew-symmetric). 4. If A is symmetric (or skew-symmetric), then kA (k is a scalar) is also symmetric for skew-symmetric matrix. 5. If A and B are symmetric matrices of the same order, then the product AB is symmetric, iff BA = AB. 6. Every square matrix can be expressed uniquely as the sum of a symmetric and a skew-symmetric matrix. 7. The matrix BT AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric matrix. 8. All positive integral powers of a symmetric matrix are symmetric. 9. All positive odd integral powers of a skew-symmetric matrix are skew-symmetric and positive even integral powers of a skew-symmetric are symmetric matrix. 8 CU IDOL SELF LEARNING MATERIAL (SLM)

10. If A and B are symmetric matrices of the same order, then (a) AB – BA is a skew-symmetric and (b) AB + BA is symmetric. 11. For a square matrix A, AAT and AT A are symmetric matrix. Conjugate of a Matrix The matrix obtained from a matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex number is called conjugate of A and is denoted by A. SOME SPECIAL TYPES OF MATRICES 1. Orthogonal Matrix Let A be a square matrix of order n, if AAT = In = AT A then A is said to be orthogonal. Properties: (i) If A is orthogonal matrix, then AAT = In = AT A is also orthogonal matrix. (ii) If A and B are any two orthogonal matrices then AB and BA is also an orthogonal matrix. (iii) If A is an orthogonal matrix then A-1 is also orthogonal matrix. 2. idempotent Matrix If a square matrix A is said to be idempotent then A2 = A. Properties: If A and B are two idempotent matrices, then ⮚ AB is idempotent, if AB = BA. ⮚ A + B is an idempotent matrix, iff AB = BA = 0 ⮚ AB = A and BA = B, then A2 = A, B2 = B 3. Unitary Matrix A square matrix A is said to be unitary, if A’A = I 4. Hermitian Matrix A square matrix A is said to be Hermitian matrix, if A = A* . Properties of Hermitian Matrix 1. If A is Hermitian matrix, then kA is also Hermitian matrix where k is any non-zero real number. 2. If A and B are Hermitian matrices of same order, then λ1A + λ2 B is also Hermitian where λ1 and λ2 are non-zero real number. 9 CU IDOL SELF LEARNING MATERIAL (SLM)

3. If A is any square matrix, then AA* and A* A are also Hermitian matrix. 4. If A and B are Hermitian matrix, then AB is also Hermitian matrix, iff AB = BA 5. If A and B are Hermitian matrix of same order, then AB + BA is also Hermitian matrix. 6. If A is a square matrix, then A + A* is also Hermitian matrix. 7. Any square matrix can be uniquely expressed as A + iB, where A and B are Hermitian matrices. 5.Skew-Hermitian Matrix A square matrix A is said to be skew-Hermitian if A* = – A or aji for every i and j. Properties: 1. If A is skew-Hermitian matrix, then kA is skew-Hermitian matrix, where k is any non-zero real number. 2. If A and B are skew-Hermitian matrix of same order, then λ1A + λ2B is also skew- Hermitian for any real number λ1 and λ2. 3. If A and B are Hermitian matrices of same order, then AB — BA is skew-Hermitian. 4. If A is any square matrix, then A — A* is a skew-Hermitian matrix. 5. Every square matrix can be uniquely expressed as the sum of a Hermitian and a skew- Hermitian matrices. 6. If A is a skew-Hermitian matrix, then A is a also Hermitian matrix. 7. If A is a skew-Hermitian matrix, then A is also skew-Hermitian matrix. 1.4 OPERATION OF MATRICES Operations on Matrices Between two or more than two matrices, the following operations are defined below: 1.4.1. Addition And Subtraction Of Matrices: Addition and subtraction of two matrices are defined in an order of both the matrices are same. (a) Addition of Matrix Let A and B be two matrices each of order m x n. Then the sum of matrices A + B is defined only if matrices A and B are of same order. If A = [aij]m×n and B = [yij]m×n, then A + B = [aij +bij]m×n, where 1 ≤ i ≤ m, 1 ≤ j ≤ n Properties of Addition of Matrices: If A, B and C are three matrices of order m x n then 10 CU IDOL SELF LEARNING MATERIAL (SLM)

1. Commutative Law A+B=B+A 2. Associative Law (A + B) + C = A + (B + C) 3. Existence of Additive Identity A zero matrix of order m x n (same as of A) is additive identity if A+0=A=0+A 4. Existence of Additive Inverse If A is a square matrix, then the matrix (- A) is called additive inverse. A + ( – A) = 0 = (- A) + A 5. Cancellation Law A + B = A + C ⇒ B = C (left cancellation law) B + A = C + A ⇒ B = C (right cancellation law) (b) Subtraction of Matrix: Let A and B be two matrices of the same order then subtraction of matrices A – B is defined as A – B = [aij – bij]n x n, where A = [aij]m x n, B = [bij]m x n If A = [aij]m×n and B = [bij]m×n, then A – B = [aij – bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n Note: (i) If A and B are matrix of order is different then A + B is not defined. (ii) Addition of matrices is an example of a binary operation on the set of matrices of the same order. 1.4.2. Multiplication Of A Matrix By Scalar Number Let A = [aij]m×n be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. if A = [aij]m×n, then kA = [kaij]m×n. Properties of Scalar Multiplication of a Matrix: Let A = [aij] and B = [bij]are two matrices of the same order m × n, then (a) k(A + B) = kA + kB, where k is a scalar. (b) (k + 2)A = k A + 2 A, where k and 2 are scalars. 11 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 1 Solution: Example: 2 If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements? Solution: Example: 3 If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements? Solution: 12 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 4 Solution: 13 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 5 Solution: Example: 6 15 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 16 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 7 Solution: 17 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 8 Solution: Example: 9 18 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 10 Solution: 19 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 11 Solution: 22 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 12 Solution: Example: 13 23 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 14 Solution: 24 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 15 Solution: 25 CU IDOL SELF LEARNING MATERIAL (SLM)

1.5 SUMMARY ● For addition and subtraction, the matrices must have the same dimensions. The resulting matrix has the same dimensions as the original matrix. ● Add each element in the first matrix to the matching element in the second matrix while performing addition. ● Subtract each member in the second matrix from the corresponding element in the first matrix while conducting subtraction. ● A positive real number multiplied by a scalar multiplies the magnitude of a real Euclidean vector without affecting its direction. 26 CU IDOL SELF LEARNING MATERIAL (SLM)

1.6 KEYWORD • Orthogonal Matrix: In a square matrix of order n, if AAT = In = AT A then A is said to be orthogonal. • idempotent Matrix: A square matrix A is said to be idempotent if A2 = A. • Hermitian Matrix :A square matrix A is said to be Hermitian matrix, if A = A* . • Symmetric matrix : A square matrix A = [aij]n x n, is said to be symmetric, if AT = A.. • Skew symmetric matrix: A square matrix A is said to be skew-symmetric matrices, if A = AT 1.7 LEARNING ACTIVITY 1. Solve for X. ___________________________________________________________________________ _______________________________________________________________  3 1   1 2 11 B  0  1 A  2 0   2 3  2. Find AB if the matrix and ___________________________________________________________________________ _______________________________________________________________  2  3 0 1 4 3 A  1 4 5 B  0  2 3 3. If 1 2 9 and 1 2 4 then find the matrix of A+B and 3A-2B. ___________________________________________________________________________ _______________________________________________________________ A   7 84 B   1 0  2 2 3 4. Find A-B if and ___________________________________________________________________________ _______________________________________________________________ 27 CU IDOL SELF LEARNING MATERIAL (SLM)

A   1 2  2 1 5. If and f(x) = (1 + x) (1 – x), then find f (a). ___________________________________________________________________________ ______________________________________________________________ 1.8 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Define a matrix. A   1 52 then find -3A. 4 2. If A   2 31 and B   1 0  2 1 3 3. If then A + B. 4. Identify the matrix  2 0  . 0 3 5. Define a scalar multiplication. Long Questions 1. Given the matrices A, B, C and D, below: 1 2 4  2 1 3  4   2 A  2 3 1 B  2 4 2 C  2 D  3  5 0 3  3 6 1  3  4  Find, if possible. i. A+B ii. C−D iii. A+D. A   3 15 and B   4 2  2 3 1 2. If then find 2A -3B.  0  2 1  2  5 6 A   4 5 2 B  4 2 1 3. Find 5A – 2B if the matrices are   3 2 1 and 8 9 1 . 28 CU IDOL SELF LEARNING MATERIAL (SLM)

 6 3 1 1 0  2 A   2 4 5 B  3 2 4  4. If the matrices are  2 0 3 and 5 3 1 then find A + 2B. A   6 34 and B   0 23 . 2 1 5. Find A +4B, given that the matrices B. Multiple Choice Questions 1.Consider A and B are two symmetric matrices of the same order, then a.AB is a symmetric matrix b.A – B is askew-symmetric matrix c. AB + BA is a symmetric matrix d. AB – BA is a symmetric matrix A  3 3 x  12  2x  x 2.If is a symmetric matrix, then x = a.4 b.3 c. 4 d.-3 3.If A is a square matrix, then A – AT is a a. diagonal matrix b. skew-symmetric matrix c. Symmetric matrix d. none of these 4 a. α = a2 + b2, β = ab b. α = a2 + b2, β = 2ab c. α = a2 + b2, β = a2 – b2 d. α = 2ab, β = a2 + b2 29 CU IDOL SELF LEARNING MATERIAL (SLM)

1 2 x 1  2 y  A  0 1 0 B  0 1 0 5.If 0 0 1 and 0 0 1  and AB = I3, then x + y equals a. 0 b. -1 c. 2 d.Nоne of these Answers 1-c 2-c, 3-b, 4-b, 5-a. 1.9 REFERENCES References book ● Vittal, P.R, “Allied Mathematics”, Reprint,Margham Publications, Chennai. ● Venkata chalapathy, S.G, “Allied Mathematics”, Margham Publications, Chennai. Textbook references ● Singaravelu, A. “Allied Mathematics”, Meenakshi Agency, Chennai. ● N. Herstein, Topics in Algebra, John Wiley and Sons, 2015. ● Gilbert Strang, Introduction to linear algebra, Fifth Edition, ANE Books, 2016 30 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 2: MATRIX 2 STRUCTURE 2.0 Learning Objectives 2.1 Introduction 2.2 Multiplication of matrices 2.3 Properties of Multiplication of Matrices 2.4 Square matrix. 2.5 Example in Multiplication of Matrices 2.6 Summary 2.7 Keywords 2.8 Learning Activity 2.9 Unit End Questions 2.10 References 2.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Describe the product of a square matrix multiplied by a column matrix, which is used to solve linear equations and describe linear transformations in linear algebra. ● Identify words with a variety of applications, including changing coordinates in geometry, solving linear equations in linear algebra, and Modeling graphs in graph theory. ● Identify one of these moves as an elementary row operation: row switching, row multiplication, or row addition. ● Describe the row space, which consists of all linear equations that can be inferred algebraically from the system's equations. 2.1 INTRODUCTION In this chapter, particularly used in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication has the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB. 31 CU IDOL SELF LEARNING MATERIAL (SLM)

In matrix multiplication was first described by the French mathematician named Jacques Philippe Marie in the year of 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is used to a basic tool of linear algebra and as such has numerous applications in many areas of applied mathematics, statistics, physics, economics, engineering and so on. Computing matrix products is a central operation in all computational applications of linear algebra. 2.2 MULTIPLICATION OF MATRICES Multiplication of Matrices: Let A and B be two matrices. If the number of columns in matrix A is equal to the number of rows in matrix B is called their product AB. 2.3 PROPERTIES OF MULTIPLICATION OF MATRICES Properties of Multiplication of Matrices: 1. Commutative Law Generally, AB ≠ BA 2. Associative Law (AB)C = A(BC) 3. Existence of multiplicative Identity AI = A = IA where I is called multiplicative Identity matrix. 4. Distributive Law A(B + C) = AB + AC 5. Cancellation Law 32 CU IDOL SELF LEARNING MATERIAL (SLM)

Let us consider A is non-singular matrix then AB = AC ⇒ B = C (left cancellation law) BA = CA ⇒B = C (right cancellation law) 6. AB = 0, does not necessarily imply that A = 0 or B = 0 or both A and B = 0 Important Points to be Remembered (i) If A and B are square matrices of the same order in n then both the product AB and BA are defined, and each is a square matrix of order n. (ii) If the matrix product AB, then the matrix A is called pre multiplier and B is called post multiplier. (iii) Generally, the rules of multiplication of matrices is row column wise (or → ↓ wise) the first row of AB is obtained by multiplying the first row of A with first, second, third, columns of B respectively; similarly second row of A with first, second, third, … columns of B, respectively and so on. 2.4 SQUARE MATRIX Definition: Square matrix is a matrix that has an equal number of rows and columns. In mathematics, m × m matrix is called the square matrix of order m. If we multiply or add any two square matrices, the order of the resulting matrix remains the same. Example:  2  9 6 3 5 4 Let A =  2 0 4 is square matrix of order 3. Properties of Multiplication Positive Integral Powers of a Square Matrix Let A be a square matrix. Then, we can define i. An + 1 = An. A, where n ∈ N. ii. Am. An = Am + n iii. (Am)n = Amn, ∀ m, n ∈ N 2.5 EXAMPLE IN MULTIPLICATION OF MATRICES Example: 1 33 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 34 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 2 Solution: 35 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 3 Solution: Example: 4 37 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 5 Solution: 39 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 6 Solution: 40 CU IDOL SELF LEARNING MATERIAL (SLM)

2.6 SUMMARY ● The identity matrix of every square matrix is a diagonal stretch of 1s from the upper- left corner to the lower-right corner, with all other members set to 0. ● Multiplying matrices is possible when inner dimensions are the same and the number of columns in the first matrix must match the number of rows in the second. ● There is no identity for non-square matrices. That is, there is no matrix such as [A][I]=[I][A]=[A] given a non-square matrix [A]. ● It is also feasible to multiply two matrices together; however, matrices can only be multiplied if the first matrix's number of columns equals the second matrix's number of rows. The two matrices can be multiplied if they meet this requirement. The outcome is a third matrix with the same number of rows as the first matrix and the same number of columns as the second matrix. 41 CU IDOL SELF LEARNING MATERIAL (SLM)

2.7 KEYWORD • Commutative : involving the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved, e.g. a × b = b × a. • Associative: involving the condition that a group of quantities connected by operators gives the same result whatever their grouping e.g. ( a × b ) × c = a × ( b × c ). • Identity : an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. • Inverse: a function or operation which reverses the order or operation of another function or operation”. • Square matrix : is a matrix with the same number of rows and columns. 2.8 LEARNING ACTIVITY 2 B   1 1  1 1 13 2 1 A  2   2 3 , write down the matrix AB. Would it be 1. If and possible to find the product of BA? If so, compute it, and if not, give reasons. ___________________________________________________________________________ _____________________________________________________________________ A   1 2  B   2 1  2 1 1 2 2. Find AB if and . ___________________________________________________________________________ _____________________________________________________________________ A   3 01 B   1 2  2 3 0 3. Check whether AB = BA if the matrices and . ___________________________________________________________________________ _____________________________________________________________________  1 2 4  2 1 0 4. Find A2 from the matrix  5 4 1 42 CU IDOL SELF LEARNING MATERIAL (SLM)

___________________________________________________________________________ _____________________________________________________________________ 5. Multiply the matrix  3 8  1 12 . 0 4 4 ___________________________________________________________________________ _____________________________________________________________________ 2.9 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Multiply  8 91 2 03 . 5 4  3 13 x    14 6 y 2. Find the value of x and y   2 1 7  and B  4 1 5 A 3 1 0 3. If possible, find BA and AB  0 2 1 A   7 52, B   0 13 . 1 2 4. Find AB if 1 B   3 1  0 11 2  1 3 A  4 2  6 1  find AB. 5. If and Long Questions 6 6 7 6 2 5 1. Find A2 from the matrix 7 5 1 .  0 2  2  6 6 7  A   2 5  2 B   6 2  5 2. Find AB if  2 2 0  and   7 5 1  . 43 CU IDOL SELF LEARNING MATERIAL (SLM)

 2 1 A  3 0   4 5 23 3. Find AB if  5 2  and B  1 0  0 6  2 A   2 2  2   2  2 0  . 4. Find A2 if Find BA if A  85 7  B   0 12  6 4 5. and B. Multiple Choice Questions 1. For any square matrix A, AAT is a a. unit matrix b.symmetric matrix c.skew-symmetric matrix d. Diagonal matrix 2. If A is any square matrix, then which of the following is skew-symmetric? a.A + AT b.A – AT c.AAT d.ATA 3. If A  13 34 and A2 – KA – 5I = 0, then K = a.5 b.3 c.7 d.8  a b c  A  b c a  c a b  where a, b, c are real positive numbers, abc = 1 and ATA = 4. If matrix I, then the value of a3 + b3 + c3 is a. 1 44 CU IDOL SELF LEARNING MATERIAL (SLM)

b. 2 c. 3 d.4 A   cos x  sin x  sin x cos x 5. If , find AAT. a. Zero Matrix b. I2 c. 11 11 d.Nоne of these Answers 1-b, 2-b, 3-a. 4-d, 5-b 2.10 REFERENCES References book ● Vittal, P.R, “Allied Mathematics”, Reprint,Margham Publications, Chennai. ● Venkata chalapathy, S.G, “Allied Mathematics”, Margham Publications, Chennai. Textbook references ● Singaravelu, A. “Allied Mathematics”, Meenakshi Agency, Chennai. ● N. Herstein, Topics in Algebra, John Wiley and Sons, 2015. ● Gilbert Strang, Introduction to linear algebra, Fifth Edition, ANE Books, 2016. 45 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 3: MATRIX3 STRUCTURE 3.0 Learning Objectives 3.1 Introduction 3.2 Rank of a matrix 3.2.1 Properties of Rank of matrix 3.3 Transpose of a matrix 3.4 Worked example 3.5 Summary 3.6 Keywords 3.7 Learning Activity 3.8 Unit End Questions 3.9 References 3.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Recognize how the order of a matrix changes when its transpose is discovered. ● Explain how to transpose matrices of different order. ● Identify symmetric and skew-symmetric matrices. ● Calculate transpose matrices and perform matrix calculations. 3.1 INTRODUCTION In this section, discuss about the systems of linear equations we mentioned that a system can have no solutions, a unique solution, or infinitely many solutions. In this section we’re going to introduce an invariant1 of matrices, and when this invariant is computed for the matrix of coefficients of a system of linear equations, it can give us information about the number of solutions to this system. We should have got knowledge of sub-matrices and minors of a matrix. Consider A be a given matrix. Matrix obtained by deleting some rows and some columns or matrix A is known as the sub-matrix of A. A matrix is called a sub-matrix of itself as it is obtained by leaving zero number of rows and columns. Minor of the matrix is the determinant of the square matrix that is obtained by deleting one row and one column from some larger matrix. 46 CU IDOL SELF LEARNING MATERIAL (SLM)

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal. i.e., it switches the row and column indices of the matrix A by producing another matrix and it is denoted by AT . In the year 1858, the transpose of a matrix was introduced by the British mathematician Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. 3.2 RANK OF MATRIX Rank of a Matrix: The maximum of number its linearly independent columns or rows of a matrix is called the rank of a matrix. The rank of a matrix cannot exceed the number of its rows or columns. Let us consider a square matrix, the columns (rows) are linearly independent only if the matrix is nonsingular. A matrix is said to be of rank zero when all of its elements become zero.The rank of the null matrix is zero.A positive integer r is said to be the rank of a non-zero matrix A, if (a) There exists at least one minor in A of order r which is not zero. (b) Every minor in A of order greater than r is zero, rank of a matrix A is denoted by ρ(A) = r. 3.2.1 Properties of Rank of a Matrix: Properties of Rank of a Matrix: 1. The rank of a null matrix is zero ie, ρ(0) = 0 2. If In is an identity matrix of order n, then ρ(In) = n. 3. (a) If a matrix A doesn’t possess any minor of order r, then ρ(A) ≥ r. (b) If at least one minor of order r of the matrix is not equal to zero, then ρ(A) ≤ r. 4. If every (r + 1)th order minor of A is zero, then any higher order – minor will also be zero. 5. If A is of order n, then for a non-singular matrix A, ρ(A) = n 6. ρ(A’)= ρ(A) 7. ρ(A*) = ρ(A) 8. ρ(A + B) = ρ(A) + ρ(B) 9. If A and B are two matrices such that the product AB is defined, then rank (AB) cannot exceed the rank of the either matrix. 47 CU IDOL SELF LEARNING MATERIAL (SLM)

10. If A and B are square matrix of same order and ρ(A) = ρ(B) = n, then p(AB)= n 11. Every skew-symmetric matrix,of odd order has rank less than its order. 12. Elementary operations do not change the rank of a matrix. Note: Echelon Form of a Matrix A non-zero matrix A is said to be in Echelon form, if A satisfies the following conditions 1. All the non-zero rows of A, if any precede the zero rows. 2. The number of zeros preceding the first non-zero element in a row is less than the number of such zeros in the successive row. 3. The first non-zero element in a row is unity. 4. The number of non-zero rows of a matrix given in the Echelon form is its rank. Example 1: Solution: Example 2: Solution: 48 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 3: Solution: Example 4: 49 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example 5: Solution: 50 CU IDOL SELF LEARNING MATERIAL (SLM)