E-LESSON-10

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B.C.A 2 All right are reserved with CU-IDOL Mathematics Course Code: BCA114 Semester: First SLM UNITS : e-Lesson No.: 10 7 www.cuidol.in Unit-10 (BCA114)

Matrix 3 33 OBJECTIVES INTRODUCTION Student will be able to : In this unit we are going to learn about the Explain Multiplication of Matrix. Matrix Multiplication. Illustrate the Square Matrix. Under this unit you will understand the square matrix. Describe the Rank of a Matrix. This Unit will also make us to understand the Rank of the matrix. www.cuidol.in Unit-10 (BCA114) INASllTITriUgThEt aOrFeDreISsTeArNveCdE AwNitDh OCNUL-IIDNOE LLEARNING

TOPICS TO BE COVERED 4 •Multiplication of Matrix •Orthogonal Matrix •Minor of Matrix •Square Matrix •Rank of Matrix •Elementary Operations www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Multiplication of Matrix 5 Multiplication of two Matrix •In order to multiply two matrices, we have to learn the method of multiplying a row by a column first. •If A is a row matrix and B is column matrix, A × B can be found provided the number of column of A = the number of rows of B. •Hence, if the order of A is 1 × m and the order of B is m × 1, only then we can find AB and the order of AB is 1 × 1. •If A = [aij] and B = [bij] are two matrix of the order m × n and n × m then the multiplication of the two matrices A and B is another matrix denoted as AB = [cij] of the order m × m. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Orthogonal Matrix 6 Orthogonal matrix: •For a orthogonal matrix if AA = I, then matrix A is called orthogonal matrix. •In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors). One way to express this is where P is the transpose of Q and I is the identity matrix. QP=I • This property of orthogonal is only applicable for the square matrix. •The orthogonal matrix has all real elements in it. •The orthogonal matrix is a symmetric matrix always. •All identity matrices are an orthogonal matrix. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Minor of Matrix 7 •In a square matrix, each element possesses its own minor. •The minor is defined as a value obtained from the determinant of a square matrix by deleting out a row and a column corresponding to the element of a matrix. • Given a square matrix A, by minor of an element [aij], we mean the value of the determinant obtained by deleting the ith row and jth column of A matrix. It is denoted by Mij. In order to find the minor of the square matrix, we have to erase out a row and a column one by one at the time and calculate their determinant, until all the minors are computed. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Minor of Matrix 8 The following are the steps to calculate minor from a matrix: •Hide ith row and jth column one by one from given matrix, where i refer to m and j refers to n that is the total number of rows and columns in matrices. •Evaluate the value of the determinant of the matrix made after hiding a row and a column from Step 1. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Square Matrix 9 •A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. •Any two square matrices of the same order can be added and multiplied. • In square matrix number of rows is equal to number of columns. • Identity matrix is always square matrix. • Diagonal always occur in square matrix. • lower and upper triangular matrix occur in square matrix. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Rank of Matrix 10 •Let A be a m × n matrix. Then, A is said to be a matrix of rank r, if: There exists at least one non-zero minor of A of order r, and Each minor of A of order greater than r is zero, i.e., each minor of order (r + 1) or higher is zero or vanishes, we generally denoted rank of matrix by (A). •Let A be a square matrix of order m × n. Then, rank of Matrix A is: (A) = n if A  0, i.e., A is non-singular. (A) < n if A = 0, i.e., A is singular. •To find the rank of a matrix by Normal form and Echelon form. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Elementary Operations 11 •There are three kinds of elementary matrix operations. 1. Interchange two rows (or columns). 2. Multiply each element in a row (or column) by a non-zero number. 3. Multiply a row (or column) by a non-zero number and add the result to another row (or column). •When these operations are performed on rows, they are called elementary row operations; and when they are performed on columns, they are called elementary column operations. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

Elementary Operations 12 How to Perform Elementary Row Operations •To perform an elementary row operation on a A, an r × c matrix, take the following steps. 1. To find E, the elementary row operator, apply the operation to an r × r identity matrix. 2. To carry out the elementary row operation, premultiply A by E. How to Perform Elementary Column Operations •To perform an elementary column operation on A, an r × c matrix, take the following steps. 1. To find E, the elementary column operator, apply the operation to an c×c identity matrix. 2. To carry out the elementary column operation, postmultiply A by E. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

MULTIPLE CHOICE QUESTIONS 13 1) State True or False: If for a square matrix A and B, null matrix O, AB =O implies BA=O: a) True b) False 2) Which of the following property of matrix multiplication is correct: a) Multiplication is not commutative in general b) Multiplication is associative c) Multiplication is distributive over addition d) All of the mentioned 3) If for a square matrix A(non-singular) and B, null matrix O, AB = O then: a) B is a null matrix b) B is a non singular matrix c) B is a identity matrix d) None of the mentioned Answers: 1) b) 2)d) 3)c) www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

MULTIPLE CHOICE QUESTIONS 14 4) The inverse of a square matrix can be found out only if it is _________. (a) Singular (b) Non-singular (c) Zero (d) Identity 5) The square matrix whose non-diagonal element are all zero and the diagonal element are all ___________. (a) Scalar (b) Square (c) Diagonal (d) None of these Answers: 4) b) 5)a) Unit-10 (BCA114) All right are reserved with CU-IDOL www.cuidol.in

SUMMARY 15 Let us recapitulate the important concepts discussed in this session: •In order to multiply two matrices, we have to learn the method of multiplying a row by a column first. If A is a row matrix and B is column matrix, A × B can be found provided the number of column of A = the number of rows of B. •In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors). One way to express this is where P is the transpose of Q and I is the identity matrix. QP=I •In a square matrix, each element possesses its own minor. The minor is defined as a value obtained from the determinant of a square matrix by deleting out a row and a column corresponding to the element of a matrix. •Each minor of A of order greater than r is zero, i.e., each minor of order (r + 1) or higher is zero or vanishes, we generally denoted rank of matrix by (A). www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

FREQUENTLY ASKED QUESTION 16 Q1. Explain multiplication of two Matrix. Ans. In order to multiply two matrices, we have to learn the method of multiplying a row by a column first. If A is a row matrix and B is column matrix, A × B can be found provided the number of column of A = the number of rows of B. For further details refer subject SLM unit 10. Q2. How to define square matrix? Ans. A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. For further details refer subject SLM unit 10. Q3. Illustrate the rank of the matrix. Ans. Let A be a m × n matrix. Then, A is said to be a matrix of rank r, if: There exists at least one non-zero minor of A of order r, and Each minor of A of order greater than r is zero, i.e., each minor of order (r + 1) or higher is zero or vanishes, we generally denoted rank of matrix by (A). For further details refer subject SLM unit 10. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

REFERENCES 17 CU-IDOL’s “Mathematics” SLM. Seymour Lipschutz, Marc Lars Lipson, “Discrete Mathematics”, Publisher: McGraw Hill Education (India) Private Limited.  J.K. Sharma, “Discrete Mathematics”, Publisher: MacMillan India Limited. K. Chandrasekhara Rao, “Discrete Mathematics”, Publisher: Narosa Publishing House. Dr. Abhilasha S. Magar, “Applied Mathematics – I”, Publisher: Himalaya Publishing House. Dr. B.S. Grewal, “Higher Engineering Mathematics”, Publisher: Khanna Publishers, Delhi. Mr. K.A. Stroud, “Engineering Mathematics”, Publisher: The MacMillan Press Ltd., London, Basingstoke. www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL

18 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-10 (BCA114) All right are reserved with CU-IDOL