BACHELOR OF COMPUTER APPLICATIONS SEMESTER - I MATHEMATICS BCA 114 First Published in 2021
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CONTENT Unit - 1: Matrix 1 4 Unit - 2: Matrix 2 31 Unit - 3: Matrix3 47 Unit - 4: Matrix4 68 Unit - 5: Matrix 5 88 Unit - 6: Linear Programming 1 111 Unit - 7: Linear Programming 2 124 Unit - 8: Different Types Of Linear Programming (L.P.) Problems 1 134 3 CU IDOL SELF LEARNING MATERIAL (SLM)
Unit - 9: Different Types Of Linear Programming (L.P.) Problems 2 150 Unit - 10: Different Types Of Linear Programming (L.P.) Problems 3 176 Unit - 11: Combinations 1 193 Unit - 12: Combinations 2 205 Unit - 13: Propositional Logic 1 217 Unit - 14: Propositional Logic 2 226 Unit - 15: Propositional Logic 3 241 4 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 5 CU IDOL SELF LEARNING MATERIAL (SLM)
● 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 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: 6 CU IDOL SELF LEARNING MATERIAL (SLM)
a11 a12 a13 ..... a1n a21 a22 a23 ..... a2 n 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: 7 CU IDOL SELF LEARNING MATERIAL (SLM)
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: 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. 8 CU IDOL SELF LEARNING MATERIAL (SLM)
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. 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. 9 CU IDOL SELF LEARNING MATERIAL (SLM)
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 10 CU IDOL SELF LEARNING MATERIAL (SLM)
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. 1 2 0 1 2 0 3 6 0 3 6 0 e.g. 0 5 9 and 0 5 9 are equal matrices, 2 6 9 4 6 9 0 8 5 0 8 6 but 2 3 4 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. 11 CU IDOL SELF LEARNING MATERIAL (SLM)
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. 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 12 CU IDOL SELF LEARNING MATERIAL (SLM)
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 13 CU IDOL SELF LEARNING MATERIAL (SLM)
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. 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. 14 CU IDOL SELF LEARNING MATERIAL (SLM)
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 1. Commutative Law 15 CU IDOL SELF LEARNING MATERIAL (SLM)
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 16 CU IDOL SELF LEARNING MATERIAL (SLM)
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. Example: 1 Solution: 17 CU IDOL SELF LEARNING MATERIAL (SLM)
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: 18 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 4 Solution: 19 CU IDOL SELF LEARNING MATERIAL (SLM)
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Example: 5 21 CU IDOL SELF LEARNING MATERIAL (SLM)
Solution: 22 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 6 Solution: 23 CU IDOL SELF LEARNING MATERIAL (SLM)
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Example: 7 25 CU IDOL SELF LEARNING MATERIAL (SLM)
Solution: 26 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 8 27 CU IDOL SELF LEARNING MATERIAL (SLM)
Solution: Example: 9 Solution: 28 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 10 Solution: 29 CU IDOL SELF LEARNING MATERIAL (SLM)
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Example: 11 Solution: 35 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 12 Solution: 36 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 13 Solution: 37 CU IDOL SELF LEARNING MATERIAL (SLM)
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Example: 14 Solution: 39 CU IDOL SELF LEARNING MATERIAL (SLM)
Example: 15 Solution: 40 CU IDOL SELF LEARNING MATERIAL (SLM)
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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. 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 42 CU IDOL SELF LEARNING MATERIAL (SLM)
1.7 LEARNING ACTIVITY 1. Solve for X. ___________________________________________________________________________ _______________________________________________________________ 3 1 Find AB if the matrix A 12 2 11 B 0 1 0 2 3 2. 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. ___________________________________________________________________________ _______________________________________________________________ 43 CU IDOL SELF LEARNING MATERIAL (SLM)
A 7 84 B 1 03 2 2 4. Find A-B if and ___________________________________________________________________________ _______________________________________________________________ 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. 2. If A 14 52 then find -3A. A 2 31 and B 1 03 then A + B. 2 1 3. If 4. Identify the matrix 2 0 . 0 3 44 CU IDOL SELF LEARNING MATERIAL (SLM)
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 . 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. 45 CU IDOL SELF LEARNING MATERIAL (SLM)
A 6 3 B 0 23 . 2 4 1 5. Find A +4B, given that the matrices and 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 1 2x x 2 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 46 CU IDOL SELF LEARNING MATERIAL (SLM)
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 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. 47 CU IDOL SELF LEARNING MATERIAL (SLM)
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 48 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. 49 CU IDOL SELF LEARNING MATERIAL (SLM)
● 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. 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. 50 CU IDOL SELF LEARNING MATERIAL (SLM)
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