E-LESSON-5 , 6

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BBA/BCOM 2 All right are reserved with CU-IDOL Business Mathematics & Statistics Course Code: BBA102/BCM 102 Semester: First SLM Unit: 5-6 E-lesson 3 www.cuidol.in Unit 5-6(BBA 102/BCM 102)

Business Mathematics & 33 Statistics OBJECTIVES INTRODUCTION To make students aware of the concept of In this unit we are going to learn about the Determinants concept of determinants To develop an understanding of various methods Under this you will learn how to solve linear to deal with linear equation using matrices and equations using methods of matrices and determinant. determinants To make students understand about the In this unit you will learn the concept of permutations and combinations concepts permutation and combination. . www.cuidol.in Unit 5-6(BBBA /1B0C2M/BC10M1)102) INSATlIlTUriTgEhOt FarDeISreTAseNrCvEedANwDithONCLUIN-IDE OLELARNING

Topics To Be Covered 4  Introduction of Basic concept of Determinants  Properties of Determinants  Cramer’s Rule  Inverse Matrix method  Introduction to permutation  Introduction to combination www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Introduction of Determinants 5 A square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aij = (i, j)th element of A. •A Determinant of a matrix represents a single number. •We obtain this value by multiplying and adding its elements in a special way. •Determinant of a Matrix of Order One Determinant of a matrix of order one A=[a11]1x1 is ������ = |a11 |= |a11 | www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Determinants of Matrices of order Three 6 ������11 ������12 ������13 • Determinant of a Matrix A = ������21 ������22 ������23 is (expanding along R1) ������31 ������32 ������33 ������11 ������12 ������13 • ������ = ������21 ������22 ������23 ������31 ������32 ������33 = (-1)1+1 ������������11 ������ ������22 ������23 - (-1)1+2������1231 ������21 ������23 + (-1)1+3������13 ������21 ������22 32 33 ������ ������ 31 32 ������ ������ 33 = ������11(������22������33 - ������32 ������23) - ������12(������21������33 - ������31 ������23) + ������13(������21������32 - ������31 ������22) = ������11������22������33 - ������11������32 ������23 - ������12������21������33 + ������12 ������31 ������23 + ������13������21������32 - ������13������31 ������22 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Examples 7 Evaluate the Determinant ∆ = 1(2) – 2(4) 12 = 2–8 42 = -6 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Evaluate the Determinant ∆ = 1 3 −4 5 8 1 −2 1 23 3 −4 5 = 3 1 −2 - (-4) 1 −2 + 5 1 1 • 1 1 −2 12 31 2 23 1 = 3(1 + 6) + 4(1 + 4) + 5(3 -2) 3 = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Properties of Determinants 9 Property 1:The value of the determinant remains unchanged if its rows and columns are interchanged. Note: det(A) = det(A’) Where A’ = transpose of A www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Property 2: If any two rows (or columns) of a determinant are 10 interchanged,then sign of determinant changes www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Property 4: If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k. 11 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Property 5:If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants. 12 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Property 6: To each element of any row or column of a determinant, the 13 equimultiples of corresponding elements of other row (or column) are added, then the value of determinant remains the same www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

3 −4 5 14 Find the minors of all elements in 1 6 −2 23 0 • M11 = Minor of a 11 = Minor of 3 = 3 6 −2 0 = (6x0) – (-2x3) = 0 + 6 = 6 • M12 = Minor of a12 = Minor of -4= 2 • M13 = Minor of a13 = Minor of 5 = 1 −2 0 = (1x0) – (-2x2) = 0 + 4 = 4 16 • M21 = Minor of a 21 = Minor of 1 = 23 = (1x3) – (2x6) = 3 -12 = -9 • M22 = Minor of a 22 = Minor of 6 = −4 5 = (-4x0) – (3x5) = 0 -15 = -15 3 0 • M23 = Minor of a 23 = Minor of -2= 2 • M31 = Minor of a 31 = Minor of 2 = 35 = (3x0) – (2x5) = 0 - 10 = -10 20 • M32 = Minor of a 32 = Minor of 3 = 1 3 −4 3 = (3x3) – (-4x2) = 9+8 = 17 • M33 = Minor of a 33 = Minor of 0 = 1 −4 5 = (-4x-2)-(6x5) =8-30 =-22 www.cuidol.in 6 −2 3 5 = (3x-2) – (1x5) = -6-5 =-11 −2 3 −4 = (3x6) – (1x-4) = 18 +4 =22 65-6(BBA Unit 102/BCM 102) All right are reserved with CU-IDOL

Adjoint of Matrices 15 • Adjoint of a matrix A = [aij]n × n is the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij . Denoted by adj A. ������11 ������12 ������13 • If A = ������21 ������22 ������23 ������31 ������32 ������33 ������11 ������12 ������13 ������11 ������21 ������31 adj A = Transpose of ������21 ������22 ������23 = ������12 ������22 ������32 ������31 ������32 ������33 ������13 ������23 ������33 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Theorem 1 : If A be any square matrix of order n, then A(adj A) = |A|In = (adj A) 16 A = lAl Verification: Let A = ������11 ������12 ������13 ������21 ������22 ������23 ������31 ������32 ������33 ������11 ������21 ������31 then adj A = ������12 ������22 ������32 ������13 ������23 ������33 Sum of the product of elements of a row (or a column) with corresponding cofactor is equal to lAl, otherwise zero. lAl 0 0 1 0 0 ∴ A (adj A) = 0 lAl 0 = lAl 0 1 0 I ..…. (i) 0 0 lAl 001 Similarly (adj A) A = lAl I …… (ii) By (i) & (ii) A(adj A) = |A|I n = (adj A) A www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

SINGULAR & NON SINGULAR 17 • Singular: A square matrix A is said to be singular if |A|=0 • Non Singular A square matrix A is said to be non-singular if |A| ≠ 0. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

18 Theorem 2: If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order. i.e. If |A|≠ 0 Then & |B|≠ 0 |AB|≠ 0 & |BA|≠ 0 Theorem 3: The determinant of the product of matrices is equal to the product of their respective determinants. i.e. |AB| =|A| |B| www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Invertible 19 Matrices If A & B are Square Matrices such that AB = BA = I B is called inverse matrix of A B = A-1 A is said to be invertible www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Theorem 4 : A square matrix A is invertible if and only if A is 20 nonsingular matrix Verification: Let A be an invertible matrix. Then there exists a matrix B such that AB = In = BA ⇒ |AB| =| In | ⇒ |A| |B| = I [ ∵ |AB| =|A| |B|] ⇒ |A| ≠ 0 ⇒ A is a non-singular matrix Conversly, let A be a non-singular matrix of order n. Then |A| ≠ 0 A(adj A) = |A|In = (adj A) A (by thm 1) ⇒A 1 adj A =I = 1 adj A A |A| |n A| ⇒ A-1 = 1 adj A |A| Hence A is an invertible matrix. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Consistency of the System of Linear Equation 21 All right are reserved with CU-IDOL Consistent system A system of equations is said to be consistent if its solution (one or more) exists. Inconsistent system A system of equations is said to be inconsistent if its solution does not exist. www.cuidol.in Unit 5-6(BBA 102/BCM 102)

Inverse of Matrices 22 If a1x+b1y+c1z = d1 a2x+b2y+c2z = d2 a3x+b3y+c3z = d3 writing these equation as AX = B ������1 ������1 ������1 ������ ������1 and B = ������2 where A = ������2 ������2 ������2 , X= ������ ������ ������3 ������3 ������3 ������3 Then X = A-1B there exists unique solution. (i)|A|≠ 0 , and (adj A)B ≠ 0, then there exists no solution. (ii)|A| = 0 and (adj A)B = 0, then system may or may not be consistent. (iii)|A| = 0 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Cramer’s Rule 23 Every square matrix (n by n) has an associated value called its determinant, shown by straight vertical brackets, such as . The determinant is a useful measure, as you will see later in this lesson. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Example 1A: Finding the Determinant of a 2 x 2 24 Matrix Find the determinant of each matrix. . = 8 – 20 =1– The determinant is –12. The determinant is . www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Check It Out! Example 1a Find the determinant of each matrix. 25 =1+9 The determinant is 10. The determinant is www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Types of Annuities 26  Annuities come in three main varieties: fixed, variable, and indexed. Each type has its own level of risk and payout potential. Fixed annuities pay out a guaranteed amount. The downside of this predictability is a relatively modest annual return, generally slightly higher than a CD from a bank.  Variable annuities provide an opportunity for a potentially higher return, accompanied by greater risk. In this case, you pick from a menu of mutual funds that go into your personal \"sub-account.\" Here, your payments in retirement are based on the performance of investments in your sub-account.  Indexed annuities fall somewhere in between when it comes to risk and potential reward. You receive a guaranteed minimum payout, although a portion of your return is tied to the performance of a market index, such as the S&P 500. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Example 2B: Using Cramer’s Rule for Two 27 Equations Step 1 Write the equations in standard form. Step 2 Find the determinant of the coefficient matrix. D = 0, so the system is either inconsistent or dependent. Check the numerators for x and y to see if either is 0. Since at least one numerator is 0, the system is dependent and has infinitely many solutions. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Permutation 28 All right are reserved with CU-IDOL • A collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter is termed as permutation. Hence, we can clearly see that the words in a language are actually some specific permutations of a collection of alphabets taken together. Now let us study more about permutations and its formula in detail below. • A language, English for example, is something that we use to communicate every day. But have you ever wondered what would happen to our understanding of the English language if the order of alphabets which form the words is changed? What if I tell you that a ‘flower’ is spelt as ‘flowre’? Would it make sense to you? No. There you have a very important example of permutation formula used in daily life. www.cuidol.in Unit 5-6(BBA 102/BCM 102)

Permutation 29 Permutation : Permutation means arrangement of things. The word arrangement is used, if the order of things is considered. Combination: Combination means selection of things. The word selection is used, when the order of things has no importance. Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, To form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of Permutation. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Permutation Now suppose that we have to make a team of 11 players out of 20 players, This is an example 30 of combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed. In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×2×1, which number is called \"n factorial\" and written \"n!\". www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Permutation 31 Number of permutations of ‘n’ different things taken ‘r’ at a time is given by:- nPr = n!/(n-r)! Proof: Say we have ‘n’ different things a1, a2……, an. Clearly the first place can be filled up in ‘n’ ways. Number of things left after filling-up the first place = n-1 So the second-place can be filled-up in (n-1) ways. Now number of things left after filling-up the first and second places = n - 2 Now the third place can be filled-up in (n-2) ways. Thus number of ways of filling-up first-place = n Number of ways of filling-up second-place = n-1 Number of ways of filling-up third-place = n-2 Number of ways of filling-up r-th place = n – (r-1) = n-r+1 www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Permutation By multiplication – rule of counting, total no. of ways of filling up, first, second -- rth-place together :- 32 n (n-1) (n-2) ------------ (n-r+1) Hence: nPr = n (n-1)(n-2) --------------(n-r+1) = [n(n-1)(n-2)----------(n-r+1)] [(n-r)(n-r-1)-----3.2.1.] / [(n-r)(n-r-1)] ----3.2.1 nPr = n!/(n-r)! Number of permutations of ‘n’ different things taken all at a time is given by:- nPn = n! Proof : Now we have ‘n’ objects, and n-places. Number of ways of filling-up first-place = n Number of ways of filling-up second-place = n-1 Number of ways of filling-up third-place = n-2 Number of ways of filling-up r-th place, i.e. last place =1 Number of ways of filling-up first, second, --- n th place = n (n-1) (n-2) ------ 2.1. nPn = n! www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Permutation 33 Examples Q. How many different signals can be made by 5 flags from 8-flags of different colours? Ans. Number of ways taking 5 flags out of 8-flage = 8P5 = 8!/(8-5)! = 8 x 7 x 6 x 5 x 4 = 6720 Q. How many words can be made by using the letters of the word “SIMPLETON” taken all at a time? Ans. There are ‘9’ different letters of the word “SIMPLETON” Number of Permutations taking all the letters at a time = 9P9 = 9! = 362880. Number of permutations of n-thing, taken all at a time, in which ‘P’ are of one type, ‘g’ of them are of second-type, ‘r’ of them are of third-type, and rest are all different is given by :- n!/p! x q! x r! Example: In how many ways can the letters of the word “Pre-University” be arranged? 13!/2! X 2! X 2! Number of permutations of n-things, taken ‘r’ at a time when each thing can be repeated r-times is given by = nr. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Combination A combination is a selection of all or part of a set of objects, without 34 regard to the order in which objects are selected. All right are reserved with CU-IDOL For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC. Computing the number of combinations. The number of Combinations of n objects taken r at a time is: nCr = n(n - 1)(n - 2) ... (n - r + 1)/r! = n! / r!(n - r)! = nPr / r! Note that AB and BA are considered to be one combination, because the order in which objects are selected does not matter. This is the key distinction between a combination and a permutation. A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged. www.cuidol.in Unit 5-6(BBA 102/BCM 102)

Combination From (1) and (2) : 35 nPr = r! . nCr or n!/(n-r)! = r! . nCr or nCr = n!/r!x(n-r)! Note: nCr = nCn-r or nCr = n!/r!x(n-r)! and nCn-r = n!/(n-r)!x(n-(n-r))! = n!/(n-r)!xr! (a) Number of combinations of ‘n’ different things taken ‘r’ at a time, when ‘p’ particular things are always included = n-pCr-p. (b) Number of combination of ‘n’ different things, taken ‘r’ at a time, when ‘p’ particular things are always to be excluded = n-pCr Example: In how many ways can a cricket-eleven be chosen out of 15 players? if (i) A particular player is always chosen, (ii) A particular is never chosen. Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL www.cuidol.in

Combination 36 Ans: (i) A particular player is always chosen, it means that 10 players are selected out of the remaining 14 players. =. Required number of ways = 14C10 = 14C4 = 14!/4!x19! = 1365 (ii) A particular players is never chosen, it means that 11 players are selected out of 14 players. => Required number of ways = 14C11 = 14!/11!x3! = 364 (iii) Number of ways of selecting zero or more things from ‘n’ different things is given by:- 2n-1 Proof: Number of ways of selecting one thing, out of n-things = nC1 Number of selecting two things, out of n-things =nC2 Number of ways of selecting three things, out of n-things =nC3 Number of ways of selecting ‘n’ things out of ‘n’ things = nCn =>Total number of ways of selecting one or more things out of n different things = nC1 + nC2 + nC3 + ------------- + nCn = (nC0 + nC1 + -----------------nCn) - nC0 Example: John has 8 friends. In how many ways can he invite one or more of them to dinner? Ans. John can select one or more than one of his 8 friends. => Required number of ways = 28 – 1= 255. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Restricted Combination (a) Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is to be always included in 37 each arrangement = r n-1 Pr-1 (b) Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is fixed: = n-1 Pr-1 (c) Number of permutations of ‘n’ things, taken ‘r’ at a time, when a particular thing is never taken: = n- 1 Pr. (d) Number of permutations of ‘n’ things, taken ‘r’ at a time, when ‘m’ specified things always come together = m! x ( n-m+1) ! (e) Number of permutations of ‘n’ things, taken all at a time, when ‘m’ specified things always come together = n ! - [ m! x (n-m+1)! ] Example: How many words can be formed with the letters of the word ‘OMEGA’ when: (i) ‘O’ and ‘A’ occupying end places. (ii) ‘E’ being always in the middle (iii) Vowels occupying odd-places (iv) Vowels being never together. Ans. (i) When ‘O’ and ‘A’ occupying end-places => M.E.G. (OA) Here (OA) are fixed, hence M, E, G can be arranged in 3! ways But (O,A) can be arranged themselves is 2! ways. => Total number of words = 3! x 2! = 12 ways. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Restricted Combination (ii) When ‘E’ is fixed in the middle 38 => O.M.(E), G.A. Hence four-letter O.M.G.A. can be arranged in 4! i.e 24 ways. (iii) Three vowels (O,E,A,) can be arranged in the odd-places (1st, 3rd and 5th) = 3! ways. And two consonants (M,G,) can be arranged in the even-place (2nd, 4th) = 2 ! ways => Total number of ways= 3! x 2! = 12 ways. (iv) Total number of words = 5! = 120! If all the vowels come together, then we have: (O.E.A.), M,G These can be arranged in 3! ways. But (O,E.A.) can be arranged themselves in 3! ways. => Number of ways, when vowels come-together = 3! x 3! = 36 ways => Number of ways, when vowels being never-together = 120-36 = 84 ways. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Restricted Combination Number of Combination of ‘n’ different things, taken ‘r’ at a time is given by:- 39 nCr= n! / r ! x (n-r)! Proof: Each combination consists of ‘r’ different things, which can be arranged among themselves in r! ways. => For one combination of ‘r’ different things, number of arrangements = r! For nCr combination number of arrangements: r nCr => Total number of permutations = r! nCr ---------------(1) But number of permutation of ‘n’ different things, taken ‘r’ at a time = nPr -------(2) From (1) and (2) : nPr = r! . nCr n!/(n-r)!x(n-(n-r))! or n!/(n-r)! = r! . nCr or nCr = n!/r!x(n-r)! Note: nCr = nCn-r or nCr = n!/r!x(n-r)! and nCn-r = = n!/(n-r)!xr! www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

SUMMARY 40  Determinant- In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|.  Permutation- A collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter is termed as permutation. Hence, we can clearly see that the words in a language are actually some specific permutations of a collection of alphabets taken together. • Combination- A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Multiple Choice Questions 41 1.There are 30 people in a group. If all shake hands with one another , how many handshakes are possible? a) 870 b) 435 c) 30! d) 29! + 1 2. In how many ways can we arrange the word ‘FUZZTONE’ so that all the vowels come together? a) 1440 b) 6 c) 2160 d) 4320 Answers: 1. b) 2. c) www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Multiple Choice Questions 42 3.In Cricket League, in first round every team plays a match with every other team. 9 teams participated in the Cricket league. How many matches were played in the first round? a) 36 b) 72 c) 9! d) 9!-1 4. How many combinations are possible while selecting four letters from the word ‘SMOKEJACK’ with the condition that ‘J’ must appear in it? rank of a matrix a) 81 b) 8!/2! c) 3!/2! d) 41 Answers: 3. a) 4. d) www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

Frequently Asked Questions 43 Q.1 What are determinants? Ans: In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Q.2 What do you understand by permutation and combination? Ans: Permutation- A collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter is termed as permutation. Hence, we can clearly see that the words in a language are actually some specific permutations of a collection of alphabets taken together. Combination- A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC. www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

REFERENCES 44 • E Knobloch, Determinants, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 766-774. • E Knobloch, From Gauss to Weierstrass : determinant theory and its historical evaluations, in The intersection of history and mathematics (Basel, 1994), 51-66. • T Muir, The Theory of Determinants in the Historical Order of Development (4 Volumes) (London, 1960). www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL

45 THANK YOU For queries Email: [email protected] www.cuidol.in Unit 5-6(BBA 102/BCM 102) All right are reserved with CU-IDOL