CU-BCA-SEM-I-Mathematics

3.6 KEYWORD • Rank of a matrix :the number of linearly independent rows or columns in the matrix • Minor: a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. • Transpose : The transpose of a matrix is obtained by changing its rows into columns and its columns into rows • Conjugate : A conjugate matrix of a matrix is obtained by replacing each term with its complex conjugate. • Transpose conjugate: If A is an m×n matrix, then the Conjugate Transpose of A is obtained by taking the complex conjugate of each entry in A and then transposing A 3.7 LEARNING ACTIVITY 1. Find the rank of matrix  5 6  . 7 8 ___________________________________________________________________________ _____________________________________________________________________  2 1 1  3 1  5 2. Find the rank of 1 1 1  . 101 CU IDOL SELF LEARNING MATERIAL (SLM)

___________________________________________________________________________ _____________________________________________________________________  1 1 1  1 1 1 3. Find the transpose of matrix  1 0 1 . ___________________________________________________________________________ _____________________________________________________________________  1 2  2  4 3 4  4. Find the rank of   2 4  4 ___________________________________________________________________________ _____________________________________________________________________ 3 1  5 1 1  2 1  5 5. Find the rank of 1 5  7 2  . ___________________________________________________________________________ _____________________________________________________________________ 3.8 UNIT END QUESTIONS A. Descriptive Questions 102 CU IDOL SELF LEARNING MATERIAL (SLM)

Short Questions 1. Find the rank of matrix 13 95 2. Find the rank on  5 7  5 7 1 2 3 2 3 4 3. Find the rank of  3 5 7 . A   2 53 B   1 0  A  BT  AT  BT 7 2 4 4. Let and verify that 5. Find the rank of 11 2 0  2 3 Long Questions 0 1 5  2 4  6 1. Find the rank of 1 1 5  . 103 CU IDOL SELF LEARNING MATERIAL (SLM)

 5 3 0   1 2  4 2. Find the rank of   2  4 8  . 1 2 1 3  2 4 1  2 3. Find the rank of  3 6 3  7 0 1 2 1 1 2 3 2 4. Find the rank of  3 1 1 3 1 1 0 1 2 3 A  2 1 3 B  2 1 3 1 2 1 and  0 1 1 Find AT and BT & verify that 5. Let A  BT  AT  BT . B. Multiple Choice Questions  0 5  7   5 0 11  1. The matrix  7 11 0  is a. A skew-symmetric matrix b. a symmetric matrix 104 CU IDOL SELF LEARNING MATERIAL (SLM)

c. a diagonal matrix d. an upper triangular matrix  5 2 x   y 2 3 2. If the matrix  4 t  7 is a symmetric matrix, then find the value of x, y and t respectively. a. 4, 2, 3 b. 4, 2, -3 c. 4, 2, -7 d. 2, 4, -7  6 8 5 A  4 2 3 3. If 9 7 1 is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is 105 CU IDOL SELF LEARNING MATERIAL (SLM)

 0 1 2 A   1 0 3 4. If   2  3 0 , then A + 2AT equals a. A b. -AT c. AT d.2A2 A   2 3 3 x  12 x x  5. If is a symmetric matrix, then x = a. 4 b. 3 c.-4 d. -3 Answers 1-a, 2-b, 3-a. 4-c, 5-c 3.9 REFERENCES References book ● Vittal, P.R, “Allied Mathematics”, Reprint,Margham Publications, Chennai. 106 CU IDOL SELF LEARNING MATERIAL (SLM)

● 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. 107 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 4: MATRIX4 STRUCTURE 4.0 Learning Objectives 4.1 Introduction 4.2 Adjoint of matrix 4.3 Inverse of a matrix 4.4 Example Problems 4.5 Summary 4.6 Keywords 4.7 Learning Activity 4.8 Unit End Questions 4.9 References 4.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Explain matrix's inverse and describing its features. ● Describe the inverse definition of a matrix commutatively, and the multiplication must operate in both directions. ● Describe a matrix must be square to be invertible, because the identity matrix must also be square. 108 CU IDOL SELF LEARNING MATERIAL (SLM)

● Demonstrate how to calculate determinants using minor and cofactor matrices. 4.1 INTRODUCTION The transpose of the cofactor matrix of that particular matrix is called theadjoint of a matrix. Consider A be given a matrix, the Adjoint is denoted by adj (A). Otherwise, the inverse of a matrix A is that matrix which when multiplied by the matrix A give an identity matrix. The inverse of a Matrix A is denoted by A-1. That is, multiplying a matrix by its inverse produces an identity matrix. Not all square matrices have an inverse matrix. If the determinant of the matrix is zero, then it will not have an inverse, and the matrix is said to be singular. Only non-singular matrices have inverses. 4.2 ADJOINT OF A MATRIX Adjoint of a Square Matrix: Consider A = [aij]m x n be a square matrix of order n and Cij be the cofactor of aij in the determinant |A| then the adjoint of A and it is denoted by adj (A), is defined as the transpose of the matrix, formed by the cofactors of the matrix. Properties of Adjoint of a Square Matrix If A and B are square matrices of order n, then (a) A (adj A) = (adj A) A = |A|I adj (A’) = (adj A)’ adj (AB) = (adj B) (adj A) adj (kA) = kn – 1(adj A), k ∈ R 109 CU IDOL SELF LEARNING MATERIAL (SLM)

adj (Am) = (adj A)m adj (adj A) = |A|n – 2 A, A is a non-singular matrix. |adj A| =|A|n – 1 ,A is a non-singular matrix. |adj (adj A)| =|A|(n – 1)2 , A is a non-singular matrix. Adjoint of a diagonal matrix is a diagonal matrix. 4.3 INVERSE OF MATRIX Inverse of a Square Matrix If A be a square matrix of order n, then a square matrix B such that AB = BA = I is called inverse of A and it is denoted by A-1. i.e., AA-1 = A-1A = 1 Properties of Inverse of a Square Matrix 1. Square matrix A is invertible if and only if |A| ≠ 0 2. (A-1)-1 = A 3. (A’)-1 = (A-1)’ 4. (AB)-1 = B-1A-1 In general (A1A1A1 … An)-1 = An-1An – -1 … A3-1A2-1A1-1 1 5. If a non-singular square matrix A is symmetric, then A-1 is also symmetric. 6. |A-1| = |A|-1 7. AA-1 = A-1A = I 110 CU IDOL SELF LEARNING MATERIAL (SLM)

8. (Ak)-1 = (A-1)Ak for k ∈ N Elementary Transformation Any one of the following operations on a matrix is called an elementary transformation. 1. Interchanging any two rows (or columns), denoted by Ri←→Rj or Ci←→Cj 2. Multiplication of the element of any row (or column) by a non-zero quantity and denotedbyRi → kRi or Ci → kCj 3. Addition of constant multiple of the elements of any row to the corresponding element of any other row, denoted byRi → Ri + kRj or Ci → Ci + kCj Equivalent Matrix ▪ Two matrices A and B are said to be equivalent, if one can be obtained from the other by a sequence of elementary transformation. ▪ The symbol≈ is used for equivalence. Relation Between Adjoint and Inverse of a Matrix Consider a matrix B, and another matrix C such that, B × C = C × B = I, then C is called as the inverse of B. When a number is multiplied by its reciprocal, then we will get 1. Same way, when we multiply a matrix by its inverse, we will get an Identity matrix. The inverse of a matrix is usually used to find the solution to a system of linear equations. Determinants and adjoints are used to find the inverse of a square matrix. The inverse of B is denoted by B-1. The relationship between the adjoint adj(B) and the inverse of a matrix B-1 is represented as B-1 = (1/|B|) × adj(B). 111 CU IDOL SELF LEARNING MATERIAL (SLM)

4.4 EXAMPLE PROBLEMS Example: 1 Solution: 112 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 2 Solution: 113 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 3 Solution: 114 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 4 Solution: 115 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 5 116 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 117 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 6 Solution: 118 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 7 Solution: Example: 8 120 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 121 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 9 Solution: 122 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 10 Solution: 123 CU IDOL SELF LEARNING MATERIAL (SLM)

Example: 11 124 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 12 125 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: Example: 13 126 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 127 CU IDOL SELF LEARNING MATERIAL (SLM)

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Example: 14 Solution: 130 CU IDOL SELF LEARNING MATERIAL (SLM)

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4.5 SUMMARY ● In a system of linear equations, a matrix is commonly employed to represent the coefficients, and the determinant can be used to solve those equations. ● If the determinant of a matrix is not zero, it has a unique inverse. ● Write the m matrix on the left and the m identity matrix on the right to determine the inverse of a m matrix: Then row reduce the matrix to get a mm identity matrix on the left, which is reduced row-echelon form. ● On the right, the new mm matrix is the multiplicative inverse of the original matrix. ● Assume A=[aij] is an n-dimensional square matrix. The adjoint of a matrix A is hence the transpose of A's cofactor matrix. 4.6 KEYWORDS • Adjoint : The adjoint of a matrix A is the transpose of the cofactor matrix of A • Inverse: The inverse of matrix is another matrix, which on multiplying with the given matrix gives the multiplicative identity. • Invertible: capable of being inverted or subjected to inversion an invertible matrix. • Singular : A matrix is said to be singular if and only if its determinant is equal to zero. • Non-singular : A non-singular matrix is a square one whose determinant is not zero 134 CU IDOL SELF LEARNING MATERIAL (SLM)

4.7 LEARNING ACTIVITY 1 0 1 3 4 5  1. Find inverse of matrix 0  6  7 ___________________________________________________________________________ _____________________________________________________________________  2 4 3 1 2 3 2. Find the determinant of  4 5 1 ___________________________________________________________________________ _____________________________________________________________________ 3. Find the inverse of matrix  1 23  0 ___________________________________________________________________________ _____________________________________________________________________  2 1 1  1 2 5 A  0 1 0  B   3 2 1 1 3 1 and  1 1 1 prove that AB1  B 1 A1 4. Let 135 CU IDOL SELF LEARNING MATERIAL (SLM)

___________________________________________________________________________ _____________________________________________________________________ 5. Find the adjoint of  2 15 . 4 ___________________________________________________________________________ _____________________________________________________________________ 4.8 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Find the determinant of  1 62 . 3  2 4 5 1 2 0 2. Find the determinant of matrix  1 4 1 3. Find the inverse of matrix  1 2  4 5 4. Find the inverse of  1 22  1 136 CU IDOL SELF LEARNING MATERIAL (SLM)

5. Define the inverse of matrix. Long Questions  1 1 2  1 2 3 1. Using elementary transformation, find the inverse of matrix  3 1 1  2 1 1   6 7  5 2. Find the inverse of matrix  5  4 3  1 2 4 5  2 0 3. Find the inverse of matrix 3 1 6  8 0 1 4 5 6  4. Find the inverse of  2 1 7  1 1 2 2 1 2 5. Find the inverse of matrix  2 1 1 B. Multiple Choice Questions 137 CU IDOL SELF LEARNING MATERIAL (SLM)

A   1 3  2 7 1. Find the inverse of the matrix , using elementary row transformation. 2. 3. 138 CU IDOL SELF LEARNING MATERIAL (SLM)

4. 5. Answers 1-a, 2-d, 3-a. 4-a, 5-d 4.9 REFERENCES References book ● Vittal, P.R, “Allied Mathematics”, Reprint,Margham Publications, Chennai. 139 CU IDOL SELF LEARNING MATERIAL (SLM)

● 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. 140 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT - 5: MATRIX 5 STRUCTURE 5.0 Learning Objectives 5.1 Introduction 5.2 Eigen values and Eigen vectors 5.2.1 Rule to find the Eigen Values and Eigen Vectors 5.3 Properties of Eigen values and Eigen vectors 5.4 Problems in Eigen values and Eigen vectors 5.5 Summary 5.6 Keywords 5.7 Learning Activity 5.8 Unit End Questions 5.9 References 5.0 LEARNING OBJECTIVES After studying this unit, you will be able to: ● Construct, and give examples of mathematical expressions that involve vectors, matrices, and linear systems of linear equations. 141 CU IDOL SELF LEARNING MATERIAL (SLM)

● Describe the mathematical expressions to compute quantities that deal with linear systems and Eigen value problems. ● Identify to mathematical statements and expressions (for example to assess whether a particular statement is accurate or to describe solutions of systems in terms of existence and uniqueness). ● Write logical progressions of precise mathematical statements to justify and communicate your reasoning. ● Apply linear algebra concepts to model, solve, and analyze real-world situations 5.1 INTRODUCTION We will study the behavior of linear endomorphisms of R-vector spaces, i.e., R-linear transformations T: V → V, by studying subspaces E ⊆ V which are preserved via scaling by the endomorphism: T(x) = λx for all x ∈ E. Such a subspace is called an Eigen space of the endomorphism T, associated to the number λ, which is called an Eigen value. A nonzero vector x such that T(x) = λx for some number λ is called an eigenvector. “Eigen-” is a German adjective which means “characteristic” or “own”. Henceforth, we’ll bandy the prefix “Eigen-” about without apology, whenever we refer to objects which arise from Eigen spaces of some linear endomorphism. 142 CU IDOL SELF LEARNING MATERIAL (SLM)

Understanding Eigen data associated to a linear endomorphism T is among the most fruitful ways to analyze linear behavior in applications. There are three primary, non-exclusive themes, which I name principal directions, characteristic dynamical modes, and spectral methods. Principal directions arise whenever an eigenvector determines a physically/geometrically relevant axis or direction. Characteristic dynamical modes arise in dynamical problems, whenever a general solution is a superposition (i.e., a linear combination) of certain characteristic solutions. Spectral methods involve studying the eigenvalues themselves, as an invariant of the object to which they are associated. 5.2 EIGEN VALUES ANDEIGEN VECTORS Definition: Let A = [aij] be a square matrix of order n. If there exists a non-zero column matrix X and a scalar λ, such that AX = λX then λ is called an Eigen value of the matrix A and X is called the Eigen vector corresponding to the Eigenvalue λ. Steps to find the Eigenvalues and the corresponding eigenvectors of a square matrix A. Let λ be an Eigenvalue of A and X be the corresponding Eigenvector. Then by Definition, AX = λX = λIX, where I is the identity matrix of order n. ⇒(A − λI) X= 0 (1) This represents a system of linear homogeneous equations in x1, x2, ….,xn. 143 CU IDOL SELF LEARNING MATERIAL (SLM)

The homogeneous system has a non-trivial solution if |A − λ I| = 0 (2) The equation |A − λ I| = 0 is called the characteristic equation of the matrix A. When we solve the characteristic equation, we get n values for λ. These n roots of the characteristic equation are called the characteristic roots or latent roots or Eigen values of A. substitute the value of λ in the equations (2), we get a non-zero (non- trivial) solution of X. X is called the invariant vector or latent vector or Eigen vector of A corresponding to the Eigen value λ. 5.2.1 Rule to Find The Eigen Values And Eigen Vectors Given a square matrix A, write the characteristic polynomial (A− λI) X= 0 Write the characteristic equation |A− λI| = 0 Solve the characteristic equation |A− λI| = 0 for the eigen values λ For each eigen value λ, solve the homogeneous system of equations (A− λI) X= 0 Note: The Eigen vector corresponding to an Eigen value is not unique. 1. If all the Eigen values λ1,λ2,…,λnof a matrix A are distinct, then the corresponding Eigenvectors are linearly independent. 2. If two or more Eigen values are equal, then it is Eigenvectors may be linearly independent or linearly dependent. 144 CU IDOL SELF LEARNING MATERIAL (SLM)

5.3 PROPERTIES OFEIGENVALUES Properties of Eigen values: • The sum of the Eigen values of a matrix A is equal to the sum of the principal diagonal elements of A.(The sum of the principal diagonal elements is called the Trace of the matrix.) • The product of the Eigen values of a matrix A is equal to |A|. • A square matrix A and its transpose AT have the same Eigen values. • If |A| = 0, i.e. A is a singular matrix, at least one of the Eigen values of A is zero and conversely. • The Eigen values of a triangular matrix are just the elements in the main diagonal. • If λ is an Eigen value of a non-singular matrix A, then λ-1 is an Eigen value of A-1. • If λ be an Eigen value of a matrix A, then λ m is an Eigen value of Am and if k is a scalar kλ is an Eigen value of kA. • The Eigen values of are also symmetric matrix (i.e.a symmetric matrix with real elements) are real. • The eigenvectors corresponding to distinct Eigen values of a real symmetric matrix are orthogonal. 5.4 PROBLEMS IN EIGEN VALUS AND EIGEN VECTORES Example: 1 145 CU IDOL SELF LEARNING MATERIAL (SLM)

A3 A  3 2 1 2 Obtain the Eigen value of where Solution: The characteristic equation of A is A  I  0 . A  3 2 1 2 Given 3 2  A  I  1 2    0  (3  )(2  )  2  0 6  2  3  2  2  0 2  5  4  0 ( 1)(  4)  0   1,4. To find the corresponding Eigen vectors, consider the equation (A – λI)X = 0 3  2 2    x1   0  1    x2    When λ = 1 we have 2 2  x1   0 1 1  x2    2x1 + 2x2 = 0 x1 + x2 = 0 146 CU IDOL SELF LEARNING MATERIAL (SLM)

both equations are same as x1 + x2 = 0. ⇒x1 = X 1  1  1 1 or  1  -x2⇒ When λ = 4 we have 1 2  x1   0  1  2 x2   -x1 + 2x2 = 0 x1- 2x2 = 0 both equations are same as x1-2x2 = 0. x1 = 2x2 X 2  2 1 Example: 2 Find the Eigen values and Eigenvectors of the matrix 1 1 3 A  1 5 1 3 1 1 Solution: The characteristic equation of A is A  I  0 λ3 - S1λ2 + S2λ-S3 =0 where S1 Sum of main Diagonal elements=1+5+1 = 7 S2  Sum of the minors of the main diagonal elements 147 CU IDOL SELF LEARNING MATERIAL (SLM)

51 1 3 11 = 1 1 + 3 1 + 1 5 = (5-1) + (1-9) + (5-1) = 0 113 1 5 1  36 S3  Determinant value of A = 3 1 1 Characteristic equation is 3  72  0  36  0 Solving the Characteristic equation, the Eigenvalues–2, 3, 6 The Eigen values of A are λ = –2, 3, 6. To find the Eigen vectors: 1  1 3 x1  0   x2  0  1 5 1    Consider the equation (A – λI)X = 0 ⇒ 3 1 1 x3 0 3 1 3 x1  0 1 1   0 7  x2   Case (i) put λ = –2. 3 1 3x3  0 3x1 + x2 + 3x3 =0----------------(1) x1 + 7x2 + x3 =0----------------(2) 3x1 + x2 + 3x3 =0-----------(3) Consider equations (1) and (2) 148 CU IDOL SELF LEARNING MATERIAL (SLM)

by using the rule of cross-multiplication, we have x1  x2  x3 x1  x2  x3 1 21 3  3 211 .  20 0 20 1   X1   0  The eigenvector corresponding to λ = – 2 is  1   2 1 3 x1  0     0  1 2 1   x2   Case (ii) put λ = 3.  3 1  2x3  0 -2x1 + x2 + 3x3 =0----------------(1) x1 + 2x2 + x3 =0----------------(2) 3x1 + x2 -3x3 =0----------------(3) Consider equations (1) and (2) and Solve by method of cross-multiplication, x1  x2  x3 we have  5 5  5 1 X 2  1  1  The eigenvector corresponding to λ = 3 is  5 1 3 x1  0   x2  0  1 1 1    Case (iii) put λ = 6.  3 1  5x3 0 149 CU IDOL SELF LEARNING MATERIAL (SLM)

-5x1 + x2 + 3x3 =0----------------(1) x1 - x2 + x3 =0----------------(2) 3x1 + x2 -5x3 =0----------------(3) Consider equations (1) and (2) and solving these equations by the rule of cross-multiplication, x1  x2  x3 we have 4 8 4 4 1 X 3  8 or 2 Thus the eigenvector corresponding to λ = 6 is 4 1 Example: 3 Find the eigen values and eigen vectors of the matrix 0 1 1 A  1 0 1 1 1 0 Solution: The characteristic equation of the matrix A is A  I  0 λ3 - S1λ2 + S2λ-S3 =0 where S1  sum of main Diagonal elements=0+0+0 = 0 S2  Sum of the minors of the main diagonal elements 150 CU IDOL SELF LEARNING MATERIAL (SLM)