BCOM,BBA_sem-1_Business, mathematics and statics u-4

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2 BUSINESS MATHEMATICS AND STATISTICS www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

3 What we are learning today ✘ Representation of Matrix ✘ Types of Matrix ✘ Operation on Matrices www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

1. 4 Representation and types www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

General form 5 www.cuidol.in 5 All right are reserved with CU-IDOL Unit-1(MAP-607)

Square Matrix 6 ✘ If in a given matrix, the . number of rows and the number of columns are equal, then the matrix is said to be a square matrix. www.cuidol.in 6 All right are reserved with CU-IDOL Unit-1(MAP-607)

Rectangular Matrix 7 ✘ If in a given matrix, the number of rows and the number of columns are not equal, then the matrix is said to be a rectangular matrix or simply matrix. www.cuidol.in 7 All right are reserved with CU-IDOL Unit-1(MAP-607)

Row Matrix 8 ✘ A matrix which contains only one row is termed as a row matrix. www.cuidol.in 8 All right are reserved with CU-IDOL Unit-1(MAP-607)

Column Matrix 9 ✘ A matrix which contains only one column is termed as a column matrix. www.cuidol.in 9 All right are reserved with CU-IDOL Unit-1(MAP-607)

Transpose of a Matrix 10 ✘ Consider a Matrix A, then the transpose of A denoted AT or At can be obtained by interchanging the rows of A as columns or columns of A as rows. www.cuidol.in 10 All right are reserved with CU-IDOL Unit-1(MAP-607)

Diagonal Matrix 11 ✘ A matrix B which satisfies the following conditions is called a diagonal matrix. 1. It should be a square matrix. 2. Each of its non−diagonal elements should be zero. www.cuidol.in 11 All right are reserved with CU-IDOL Unit-1(MAP-607)

12 ✘ A is a 2 × 2 diagonal matrix and B is a 3 × 3 diagonal matrix. ✘ If A is a diagonal matrix then transpose of A is A itself, i.e., AT = A. www.cuidol.in 12 All right are reserved with CU-IDOL Unit-1(MAP-607)

Scalar Matrix 13 ✘ A Matrix B is said to be a scalar matrix, if it satisfies the following conditions: 1. It should be a diagonal matrix. 2. All the diagonal elements are one and the same. www.cuidol.in 13 All right are reserved with CU-IDOL Unit-1(MAP-607)

Identity Matrix 14 ✘ A Matrix B is said to be an identity matrix, if it satisfies the following conditions. 1. It should be a scalar matrix. 2. All the diagonal elements are equal to 1. ✘ It is denoted by the letter I. It is otherwise termed as unit matrix. www.cuidol.in 14 All right are reserved with CU-IDOL Unit-1(MAP-607)

Null Matrix 15 ✘ A matrix B is said to be a null matrix, if all of its elements equal to “0”. www.cuidol.in 15 All right are reserved with CU-IDOL Unit-1(MAP-607)

Upper Triangular Matrix 16 ✘ A Matrix B is said to be an upper triangular matrix if it satisfies the following conditions: 1. It should be a square matrix. 2. All the elements which lie below the diagonal should be zero. www.cuidol.in 16 All right are reserved with CU-IDOL Unit-1(MAP-607)

Lower Triangular Matrix 17 ✘ A Matrix B is said to be a lower triangular matrix if it satisfies the following conditions: 1. It should be a square matrix. 2. All the elements which lie above the diagonal should be zero. www.cuidol.in 17 All right are reserved with CU-IDOL Unit-1(MAP-607)

Triangular Matrix 18 ✘ A Matrix B which is both upper triangular and lower triangular is said to be a triangular matrix. www.cuidol.in 18 All right are reserved with CU-IDOL Unit-1(MAP-607)

19 OPERATIONS ON MATRICES www.cuidol.in 19 All right are reserved with CU-IDOL Unit-1(MAP-607)

20 1. Addition 2. Subtraction 3. Multiplication www.cuidol.in 20 All right are reserved with CU-IDOL Unit-1(MAP-607)

Rules for Matrices Operation 21 ✘ If A and B are any two matrices then, 1. (A + B) is possible if and only if both the Matrices A and B are of the same order. 2. (A − B) is possible if and only if both the Matrices A and B are of the same order. 3. (A × B) is possible if and only if the number of columns of the Matrix A and the number of rows of the Matrix B are equal. www.cuidol.in 21 All right are reserved with CU-IDOL Unit-1(MAP-607)

Properties of Matrix Addition 22 ✘ If A, B and C are any three matrices of the same order then, 1. Matrix addition is commutative: A + B = B + A. 2. Matrix addition is associative: (A + B) + C = A + (B + C). www.cuidol.in 22 All right are reserved with CU-IDOL Unit-1(MAP-607)

23 3. Additive identity exists: A + O = O + A = A, where O is the null matrix. 4. Additive inverse exists: A + D = D + A = O, where D is referred to as the additive inverse of A. www.cuidol.in 23 All right are reserved with CU-IDOL Unit-1(MAP-607)

Properties of Matrix Multiplication 24 ✘ If A, B and C are any three matrices then, 1. Matrix multiplication is not commutative. If AB and BA are defined, it is not necessary that AB = BA. 2. Matrix multiplication is associative. If BC, AB, A(BC) and (AB)C are defined, then A(BC) = (AB)C. www.cuidol.in 24 All right are reserved with CU-IDOL Unit-1(MAP-607)

25 3.Matrix multiplication is distributive. If B + C, A(B + C), AB and AC are defined then, A(B + C) = AB + AC 4. If A and B are any two matrices and AB is defined, then AB = O. It does not imply that either A = O or B = O where O is the null matrix. www.cuidol.in 25 All right are reserved with CU-IDOL Unit-1(MAP-607)

26 “Examples” www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

Example 1. The marks secured by x and y on two different subjects 27 and two different terms are given ✘ Find the total marks secured by them in both the terms put together. www.cuidol.in 27 All right are reserved with CU-IDOL Unit-1(MAP-607)

rLeestpTe1c,tTiv2ealyn.dTThesntand for Term 1 and Term 2 and total marks 28 ✘ Hence, the total marks secured by both of them are www.cuidol.in 28 All right are reserved with CU-IDOL Unit-1(MAP-607)

29 Example 2 q www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

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33 NOTE: B2 × 3 × A2 × 2 is not defined. Since the number of columns of the 1st matrix is not equal to the number of rows of the 2nd matrix, the matrix multiplication is not defined. www.cuidol.in 33 All right are reserved with CU-IDOL Unit-1(MAP-607)

34 Example 3 www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL

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✘ Given (1) 36 ✘ f(x) = x2 − 3x + 6 ✘ To find f(A) ✘ Since A is a matrix, when we replace X as A in Eq. (1) becomes a function in matrices. During this conversion, pure constant term should be multiplied with the relevant identity matrix. ✘ In Eq. (1), the third term is 6, which is a constant term. So, this term should be multiplied with I3 × 3. Hence, f(A) = A2 − 3A + 6I. ✘ =A×A−3×A+6×I www.cuidol.in 36 All right are reserved with CU-IDOL Unit-1(MAP-607)

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✘ A manufacturer produces three products A, B and C which he 38 sells in two markets X and Y. Annual sale volumes are indicated below: www.cuidol.in 38 All right are reserved with CU-IDOL Unit-1(MAP-607)

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✘ Then the total revenue can be expressed in a matrix 40 form, www.cuidol.in 40 All right are reserved with CU-IDOL Unit-1(MAP-607)

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42 THANK YOU www.cuidol.in Unit-1(MAP-607) All right are reserved with CU-IDOL