SLM_(1)

46 Business Mathematics and Statistics A  (1 i)n 1 Also P = n  (1 i)  1 i 1  A  (1 i) n 1 = n i  (1 i)  A 1  = 1 n i  (1 i)  A ?P=  n … (5) (1 i) 1 i  Example 3: Find the present value of an annuity of ` 300 per annum for 5 years at 4%. Answer to the nearest paisa. Given that log 1.04 = .0170333 and log 821.923 = 2.9148335. If the above annuity be a perpetual one, what will be the present value? [I.C.W.A.] Solution: Using formula (5) A P = i [1 – (1 + i)–n] 300 = .04 [1 – (1.04)–5]

Now (1.04)–5 =Antilog {–5 log (1.04)} = Antilog {–5 × .0170333} = Antilog {–.0851665} = Antilog {1.9148335} = .821923 300 P = .04 (1 – (.821923) 300 B .04 × .178077 53.423100 = .04 5342.31 = 4 CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 47 (f) 1335.5775 (g) 1335.5775 2 Present value is ` 1335.58 If the above annuity be a perpetual one, then putting n Ÿf in formula (5) we get A P= i 300 P = .04 30000 = 4 = ` 7,500 3.4 Perpetuity The present value of a perpetuity is obtained by making n infinite. A ?P= i … (6) In particular if xA is the present value of annuity A, then the annuity is said to be worth “x years’ purchase”.

A Here xA = i 3. 100 1 r r  ?x= … (6) = … (7)   i 100  100 The No. of years purchase = Rate per cent Therefore, the number of years’ purchase of a perpetual annuity is obtained when we divide 100 by the rate percent. A freehold estate yields a perpetual annuity. This is called the rent. In order to know the rate per cent at which interest is reckoned, we divide 100 by the number of years’ purchase. CU IDOL SELF LEARNING MATERIAL (SLM)

48 Business Mathematics and Statistics Example 4: A freehold estate is purchased for ` 27,500. Find the rent for which it should be let so that the owner may receive 4% on the purchase money. Solution: The present value of perpetual annuity of amount A is ` 27,500, interest being 4%. r 41 i = 100 100 25 Present value of a perpetuity is given by A P= i A 1 ? 27500 =  = 25A  25  27500 A = 25 = 1100 Therefore rent should be ` 1,100. Example 5: A freehold estate worth ` 1,200 a year is sold for ` 40,000. What is the rate of interest? Solution: Using the formula — A P=

i 1200 40,000 = i 1200 r i = 40000 100 120000 r = 40000 = 3 The rate of interest is 3%. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 49 3.5 Annuity Due To find the amount of an annuity due of ` A at the end of n years. As the first payment is made immediately, it earns interest for n years.  r n ? M = A 1  1  100 = A(1 + i)n i.e., M = ARn. 1 This is the amount of the first payment at the end of n years. The second payment which is made at the end of the first year, earns interest for (n-1) years. M = ARn–l. 2 This is the amount of the second payment. Similarly, M = ARn–2 is the amount of the third payment. Proceeding in this manner we get the amount 3 of the last payment as M3=AR ? the total amount of the annuity due is M1 + M2 + M3 + …… + Mn

? M = M1 + M2 + M3 + … + Mn = ARn + ARn–1 + ARn–2 + … + AR = AR (Rn–1 + Rn–2 + …… + 1) = AR (1 + R + R2 + ……+Rn–1)  Rn 1 ?M=AR  … (8)   … (9) R1  (1 i) n 1 = A(1 + i)    (1 i) 1  ? M = A(11) [1 1) n 1] i CU IDOL SELF LEARNING MATERIAL (SLM)

50 Business Mathematics and Statistics 3.6 Present Value of an Annuity Due To find the present value of an annuity due of Rupees A (say), for n years with an effective rate of interest r per cent per annum. Suppose P1 is the present value of the first payment of Rupees A. Since the first payment is made at the beginning of the first year, there is no interest on it. ? P1= A If P2 is the present value of the second payment, then interest is calculated for one year. A= P2R A i.e., P = = AF–1 = AV 2R A –2 2 Similarly P3 = 2 = AR = AV R Thus the present value of the last payment is A Pn = n 1 = AR–n+1 = AVn–1 R The present value of the annuity is P = P1 + P2 + P3 + …… + Pn 4. A + AV + AV2 + …… + AVn–1

5. A(1 + V + V2 + …… Vn–1) P =  1 Vn  AA  … (10) 1  V Also P =  1 Rn  A   since V = R–1 1R1   1 =1  Rn  A 1  1   R  Rn 1 R  A   =A   Rn R1  CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 51 A  R n 1 i.e., P = n1   … (11) R  … (12) … (13) ? R1  (1 i) n 1 Also, P = n1   (1 i)  1 i 1  A  (1 i) n 1 P=  n1  i  (1 i)  Formula (12) can also be written as A P = i [(1 + i) – (1 + i)–n-1)] Example 6: A man purchases a machine on instalment basis. As per the agreement he is required to pay ` 1,000 every year for 16 years and the first instalment is to be paid immediately. If the rate of compound interest works out to be 5% per annum, what is the present worth of the agreement? Solution: Using formula (13) A P = {(1 + i) – (1 + i) –(n–1)} i = 1000 {1.05 – (1.05)–16–1)}

.05 1000 = .05 {1.05 – (1.05)–15} Now (1.05)–15 = Antilog {–15 log (1.05)} = Antilog {–15 × .0212} = Antilog {–.3180} = Antilog (1.6820) = 4808 1000 P=  5   (1.05 – .4808)  100  CU IDOL SELF LEARNING MATERIAL (SLM)

52 Business Mathematics and Statistics 100000 =  .5692 5 100000 5692 = . 5 10000 = ` 11,384 The present worth of the agreement is ` 11,384. 3.7 Deferred Annuity Amount of a deferred annuity: To find the amount of an annuity of Rupees A for n years, which is deferred for m years, the effective rate of interest being r per cent per annum. As the amount is deferred for m years, the first payment falls due at the end of (m + 1) years, and there is no interest for the period of m years. The amount at the end of (m + n) years would be the same as the amount of immediate annuity of n years. A M = i [(1 + i)n – 1] 3.8 Present Value of a Deferred Annuity

To find the present value of an annuity of Rupees A for n years, deferred for m years. The effective rate of interest is r per cent per annum. The first payment of ` A falls due at the end of (m + 1) years. Therefore the present value of annuity of ` A would be: A = Rm 1 because A = P1Rm+1, where P1 is the present value. Thus A A m1 P = m1 m1 AV 1R (1 i) CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 53 If P2 is the present value of the second payment of ` A which falls due at the end of (m + 2) years, then A = P2 Rm+2 A A m2 ?P = m2 m2 AV 2R (1 i) Proceeding in this manner, A A m2 P = mn m2 AV nR (1 i) where Pn is the present value of the last payment of ` A which falls due at the end of (m + n) years. Thus, the present value of the annuity is: P = P1 + P2 + P3+ ……Pn = AVm+1+ AVm+2 + ……+AVm+n = AVm+1 (1 + V + ……Vn–1) 1 Vn  = AVm+1     = Rm 1  1V   1  A  1 Rn  A

= m1    R  Rn   R1  ?  Rn 1 ?P=  R 1  Rm n  =  (1 i)n 1  Also, P =  mn  (1 i 1)  (1 i)  A  (1 i) n 1 i.e., P =  mn  i  (1 i)  … (14) … (15) CU IDOL SELF LEARNING MATERIAL (SLM)

54 Business Mathematics and Statistics Example 7: What is the present value of an annuity of ` 100 payable for 12 years but deferred for 3 years, the effective rate of interest being 5% p.a.? Solution: Using formula (15), A  (1 i) n 1 P=  mn  i  (1 i)  (1.05)12 1 =  (1.05)15  .05   100 =  5  [(1.05)12 – 1] × (1.05)–15  100 10000 = 5 (1.805 – 1) × 0.4808 10000 = (1.805 – 1) × 0.4808 5 10000 805 4808 =  5 1000 10000 = ` 774.088 ? The present value is ` 774.09 approx.

3.9 Endowment Fund This fund is created for the payment of an annuity for a definite period. Therefore the formula for this fund is the same as the present value of an annuity. This fund is invested for a certain period at compound interest. Further, this fund being the same as the present value of a deferred perpetuity, we have, A1 … (16) P = i (1 i)n where n = number of years for which it is deferred. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 55 Example 8: A person wishes to institute a scholarship of the value of ` 1200 to be awarded every year to a deserving student. Assuming that money is worth 5% p.a. compound interest, find the amount he should provide. Solution: Here the formula for the present value of a perpetuity is applicable. Therefore A P= i 1200 1200 .05  1 = = 24,000    20 The person concerned has to provide ` 24,000. Example 9: Find the present value of a perpetual annuity of ` 800, the first payment of which is to be made in six years time, interest being reckoned at 4% per annum. Answer to the nearest paisa. Solution: Using the formula (16) A1 P = i (1 i)n and substituting A = 800 i = .04 n = 6–1=5

we get 800  (1.04)5 P= .04 z Antilog [log 800-5 log (l.04)–log (.04)] z Antilog [2.9031 – 5(0.170) – 2.6021] z Antilog (4.2160) z 16440 The present value is ` 16,440. CU IDOL SELF LEARNING MATERIAL (SLM)

56 Business Mathematics and Statistics 3.10 Annuity in the case of which payments are made other than annually If an annuity A is payable t times every year and interest is compounded t times a year, then the value of each payment is A/t. The ‘interest periods’ in n years is nt. The interest on unit sum for an interest period (1/t) is = 1   were i is the rate per cent per annum. t 100  Example 10: What will an annuity of ` 400 (annual value) amount to in 4 years at 4% compound interest, payable quarterly? Solution: Using the formula: A M = i {(1 + i)n – 1} 4 4  400  .04   1  1 =  .04  4      4 400  =  1  1.01)16 1

   100 = 100 × 400 {(1.01)16 – 1} Now (1.01)16 = Antilog {16 log 1.01} = Antilog {16 (.0043)} = Antilog {0.0688} = 1.171 M = 100 × 400 (1. 171 – 1) = 100×400× 171 40000 1000 = 1000 × 171 The amount is ` 6840. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 57 3.11 Sinking Fund This fund is created for the purpose of discharging a known liability or for replacing a wasting asset. Further, it is very often created for redemption of debentures. If M is the amount of liability to be discharged after n years and A is the sum invested at the end of every year, then A M = i {(1 + i)n – 1} Example 11: A man wishes to create a fund of ` 50,000 for a house at the time of retirement which is due after 10 years. If the rate of compound interest is 5% per annum, find the amount that he should deposit each year to receive the fund at the time of his retirement. A Solution: Here M = i {(l + i)n – 1} is the formula to be used. A ? 50,000 = .05 {(1.05)10 – 1} A =  1  (1.6289 – 1)   20 = 20A × (0.6289)

?A= 50000 20  0.06289 2500 ? 0.6289 ? Antilog (log 2500 – log 0.6289) ? Antilog (3.3979 – 1.7986) ? Antilog (3.5993) = 3975 The man has to deposit ` 3,975 every year. 3.12 Repayment of Loan by Instalments In problems relating to repayment of loan by instalments, we should use the formula for the present value of an annuity. In problems connected with investment or saving by instalment or with CU IDOL SELF LEARNING MATERIAL (SLM)

58 Business Mathematics and Statistics the creation of ‘Depreciation Fund’ by instalment, we should use the formula for the amount of annuity. Example 12: A man wishes to pay back his debt of ` 2,522 due after 3 years by three equal yearly instalments. Find the amount of each instalment, money being worth 5% per annum compound interest. Solution: Here we use the formula for the amount of annuity since the debt is due after 3 years. A M = i {(1 + i)n – 1} A = .05 {(1.05)3 – 1} A ? 2522 = .05 (1.05 – 1) {(1.05)2 + 1.05 + 1} = A (3.1525) 2522 ? A = 3.1525 =`800 ` The amount of each instalment is ` 800. Additional Worked Examples Example 13: A man purchases a house and takes a mortgage on it for ` 60,000 to be paid off in 12 years by equal annual payments. If the interest rate is 5% compounded annually, what amount will he be required to pay each year?

Solution: Using formula (5), i.e., A P = i {1 – (1 + i)–n} we get A 60,000 = .05 {1 – (1.05)–12} A n  1 {1 – (1.05)–12}  20 CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 59 Now (1.05)–12 =Antilog {–12 log (1.05)} l = Antilog {–12 × .02119} i = Antilog {–0.25428} = Antilog ( 1 .74572) = 0.55683 ? 60,000 = 20A (l – 0.55683) i.e., 60,000 = 20A × 0.44317 = (8.8634)A 60,000 A = 8.8634 60,000* [ 8.8634 = Antilog [log 60,000 – log 8.8634] = Antilog [4.77815 – 0.94760] = Antilog (3.83055) = 6,769.49 = ` 6,769.49 Each year the person concerned has to pay ` 6,769.49. Example 14: A penon purchased a refrigerator and paid ` 5,000 immediately. He cleared off his debt completely by paying ` 5,000 after one year and another sum of ` 5,000 after two years. If he paid compound interest at 3Y2% per annum, find the selling price of the refrigerator to the nearest rupee. Solution: Using formula (5)

A P = i {1 – (1 + i)–1} we get  2 5000  3½  1 1   =  3½    100       100   2 5000  207   1   =  7   100     200  CU IDOL SELF LEARNING MATERIAL (SLM)

60 Business Mathematics and Statistics 10,00,000  40,000  = 1  7  42,849  10,00,000 2,849 = 7 42,849 = ` 9,498.47 Since ` 5,000 was paid immediately, the selling price of the refrigerator is 5 5,000 + ` 9,498.47 = ` 14,498.47 Example 15: A man borrows ` 5,000 at 4% compound interest. If the principal and interest are to be repaid by 10 equal annual instalments, what is the amount of each instalment? It is given that Log 1.04 = .0170333 and log 675565 = 5.829667 Solution: Using formula (5) A i.e., P = i {1 – (I + i)–n} we get A 5000 = .04 {1 – (1.04)–10} Now (1.04)–10 = Antilog {–10 log (1.04)}

= Antilog {–10 (.0170333)} = Antilog {–.170333} = Antilog ( 1 .829667) = .675565 A 5000 = .04 (l – 0.675565) A = .04 (0.324435) 5000  .04 A= 0.324435 = ` 616.4561 The amount of each instalment is ` 616.46 approximately. CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 61 3.13 Summary ‘Annuity’ is a fixed sum of money paid regularly at equal intervals of time. Various types of Annuities are: Immediate Annuity, Annuity Due, and Deferred Annuity. Endowment Fund is created for the payment of Annuity for a definite time period. Sinking Fund is created for the purpose of discharging a known liability or for replacing a wasting asset. Repayment of loans by instalments is based on the basis of calculation of annuity. So, also the examples on ‘Depreciation Fund’. 3.14 Key Words/Abbreviations I.A. = Immediate Annuity, A.D. = Annuity Due, D.A. = Deferred Annuity, P.W. = Present Worth, E.F. Endowment Fund, S.F. = Sinking Fund, D.F. = Depreciatioin Fund 3.15 Learning Activity Study the illustrative examples that explain the calculation of I.A., A.D., D.A., P.W., E.P., S.F., D.F.,M.S. Study the importance of each of the funds. .............................................................................................................................................................. ...........................................................................................................................................................

3.16 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions = A person borrows ` 6000 at 5% compound interest and agrees to pay both the principal and the interest in 5 equal annual instalments at the end of each year. Find the amount of each instalment. [Ans: ` 1,385.85] = A person deposits his whole fortune of ` 20,000 in a bank at 5% compound interest per annum and settles to withdraw ` 1,800 per year for his personal expenses. If he begins to spend from the end of the first year and goes on spending at this rate, show that he will be ruined before the end of the 17th year. [I.C.W.A.] CU IDOL SELF LEARNING MATERIAL (SLM)

62 Business Mathematics and Statistics ` A man procures a machine for ` 1,750. He agrees to pay ` 800 cash immediately and the balance in 9 equal annual instalments. If interest (compound) is calculated at 4½% p.a., find the amount of each instalment. [Ans: ` 130.87] 4. How many year’s purchase should be given for a free hold estate, interest being calculated at 4½%? [Ans: 222/ ] 9 5. A sinking fund is created for the redemption of debentures of ` 1,00,000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund, if the investment can earn interest @ 4% per annum? [Ans: ` 2,408.19] [Burdwan Uni.] ? A sinking fund is to be created for the purpose of replacing, after 20 years, some machinery worth ` 1,00,000. How much money should be set aside each year out of the profits for the sinking fund, if the rate of compound interest is 5% p.a.?[Ans: ` 3021.15] [Burdwan Uni.] ? A person borrows ` 4,000 on the condition that he will repay the money with compound interest at 5% p.a. in 6 equal annual instalments, the first one being payable at the end of the first year; find the value of each instalment. [Ans: ` 787.71] [Burdwan Uni.] = A man purchases a house and takes a mortgage on it for ` 60,000 to be paid off in 12 years by equal annual payments. If the interest rate is 5 per cent compounded annually, what amount will he be required to pay each year? [Ans: ` 6,769.41] [I.C.W.A.] ` A person proposes to make an endowment on 1st July 1999 to the University by depositing a sum of money to its banking account stipulating: (i) the payment of a scholarship of ` 1,000 per annum for 10 years, and (ii) the award of book prizes of the value of ` 500 every year for 20 years. The money deposited together with compound interest @ 5% per annum is to exhaust itself at the end of the said 20 years. Assuming that the payments and awards

stipulated are to be made on 1st July every year commencing with the year 2000, determine the amount of endowment required to be made on 1st July 1999 for this purpose. [Ans: ` 13,957] [Calcutta Uni.] 10. Find the present value of a perpetual annuity of ` 10 payable at the end of the first year, ` 20 at the end of the second year, ` 30 at the end of the third year and so on and in this manner increasing ` 10 each year; interest being taken at 5% per annum. [Ans: ` 4,200] CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 63 = A man retires at the age of 60 years and his employer gives him a pension of ` 1,200 a year paid in half-yearly instalrnents for the rest of his life. Reckoning his expectation of life to be 13 years and that interest is at 4% p.a. payable half-yearly, what single sum is equivalent to this pension? [Ans: ` 12,075] [Calcutta Uni.] = The population of a country increases every year by 2.4% of the population at the beginning of that year. In what time will the population double itself? Answer to the nearest year. [Ans: 29 years (approx)] [I.C.W.A.I] ` A man wishes to have ` 2,500.00 available in a bank account when his daughter’s first year college expenses begin. How much must he deposit now at 3.5% compounded annually, if the girl is to start in college six years hence? [Ans: ` 2,034] [I.C.W.A.I.] ? When a boy is born, ` 500.00 is placed to his credit in an account that pays (i) 6% compounded annually, (ii) 6% compounded quarterly, (iii) 6% compounded monthly. If the account is not disturbed, what amount will there be to his credit on his twentieth birthday? [Ans: (i) ` 1,603 (ii) ` 1,626 (iii) ` 1,596] [I.C.W.A.I.] ` A machine depreciates in value each year at the rate of 10% of its value at the beginning of a year. The machine was purchased for ` 10,000. Obtain, to the nearest rupee, its value at the end of the tenth year. [Ans: ` 3,483] [I.C.W.A.I.] = A machine depreciates at the rate of 7% of its value at the beginning of a year. If the machine was purchased for ` 8500, what is the minimum number of complete years at the end of which the worth of the machine will be less than or equal to half of its original cost price? [Ans: 10 years] [I.C.W.A.I.]

= A truck purchased by a transport company at ` 60,000 depreciates at the rate of 10% p.a. and its maintenance cost for the first year is ` 2,000 which increases by 2% every year. If the scrap value realized when sold is ` 35,429.40, find the minimum average annual return from the truck the company should get so as not to sustain any loss. [Ans: ` 6,995.74] [I.C.W.A.I.] CU IDOL SELF LEARNING MATERIAL (SLM)

64 Business Mathematics and Statistics = A sum of money invested at compound interest amounts to ` 21,632 at the end of the second year and to ` 22,497.28 at the end of the third year. Find the rate of interest and the sum invested. [Ans: 4% p.a., ` 20,000] [ I.C.W.A.I.] ` A man wants to invest ` 5,000 for four years. He may invest the amount at 10% per annum compound interest, interest accruing at the end of each quarter of the year or he may invest it at 10½% per annum compound interest, interest accruing at the end of each year. Which investment will give him slightly better return? [Ans: 2nd investment will give him slightly better return] [I.C.W.A.I.] + Mr. Brown was given the choice of two payment plans on a piece of property. He may pay ` 10,000 at the end of 4 years, or ` 12,000 at the end of 9 years. Assuming money can be invested annually at 4% per year converted annually, what plan should Mr. Brown choose? [Ans: 2nd Plan] [ I.C.W.A.I] 21. The population of developing countries increases every year by 2.3% of the population at the beginning of that year. In what time will the population double itself? Answer to the nearest year. [Ans: 31 years] [I.C.W.A.I.] ? In a certain population the annual birth and death rates per 1000 are 39.4 and 19.4 respectively. Find the number of years in which the population will be doubled assuming that there is no immigration or emigration. [Ans: 35 years] [I.C.W.A.I.] = To accumulate a fund for his son’s higher education a person invests a sum of ` 100 on the son’s first birthday and an equal amount on each of the subsequent birthdays. If the interest is compounded half yearly at the rate of 6% per annum, find the amount accumulated just after the investment has been made on the l8th birthday. [Ans: ` 2,341.44]

= A person borrows a sum of ` 5,000 at 4% compound interest. If the principal and interest are to be repaid in 10 equal annual instalments, find the amount of each instalment, first payment being made after 1 year. [Ans: ` 616.52] [I.C.W.A.] ? A company sets aside as reserve the sum of ` 20,000 annually to enable it to payoff a debenture issue of ` 2,39,000 at the end of 10 years. Assuming that the reserve accumulates at 4% per annum compound interest, find the surplus after paying off the debenture stock. [Ans: ` 1,122.14] [l.C.W.A.] CU IDOL SELF LEARNING MATERIAL (SLM)

Annuities 65 B. Multiple Choice/Objective Type Questions ? What is the amount to be set aside every year so as to yield ` 175 at the end of 8 years at 5% C.I. p.a.? (a) ` 18.32 (b) ` 24.50 (c) ` 28 (d) ` 30 None of the above ? What is the present value of an immediate annuity of ` 150 per year for 10 years at 3% p.a. compound interest? (a) ` 11,050 (b) ` 1,279.53 (c) ` 1,830 (d) ` 1,980 = None of the above r A person borrows ` 6,000 at 5% C.I., p.a. and agrees to pay both the priincipal and interest in 5 equal annual instalments at the end of each year. (a) ` 1,519.26 (b) ` 1,250.40 (c) ` 1,385.68 (d) ` 1,630.90 (e) None of these. Answers (1) (a); (2) (b); (3) (c) 3.19 References References of this unit have been given at the end of the book.

ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

66 Business Mathematics and Statistics UNIT 4 MATRICES Structure 4.0 Learning Objectives 4.1 Introduction 4.2 Different Types of Matrices 4.3 Addition of Matrices 4.4 Multiplication of Matrices 4.5 Application of Matrices 4.6 Orthogonal Matrix 4.7 Inverse of a Matrices

4.8 Cofactor Matrix 4.9 Adjoint Matrix 4.10 Method of Solving Simultaneous Equations Using Matrix Property 4.11 Summary 4.12 Key Words/Abbreviations 4.13 Learning Activity 4.14 Unit End Questions (MCQ and Descriptive) 4.14 References CU IDOL SELF LEARNING MATERIAL (SLM)

Matrices 67 4.0 Learning Objectives After studying this unit, you will be able to:  Explain the meaning and definition of a ‘Matrix’ and the various types of matrices.   Learn ‘Matix Operations of Addition and Multiplication’ along with the illustrative exmples.   Elaborate the Orthogonal Matrix, inverse of a Matrix, Co-factor Matrix, Adjoint Matrix and the method of calculation through illustrative examples.  Illustrate the method of solving simultaneous equations using Matrix Property. 4.1 Introduction A matrix is a rectangular array of m rows and n columns arranged within two brackets as shown below: xxx x  11 12 13 .......... 1n   x x x .......... x   21 22 23 2n   ..... ..... ..... .......... .....    ..... ..... ..... .......... .....   ..... ..... ..... .......... .....  x x x x  m1 m2 m3 .......... mn 

Capital letters such as A,B, C .......... can be used to denote matrices. The above matrix becomes a square matrix of order n if m = n Further, 2 1 4 9 3 1 2 6 If A =   3 4 5 6 Then it is said to be a matrix of order 3 × 4. 3 4 4 4   2 6  is a square matrix of order 3. Also B =  2 3 8 CU IDOL SELF LEARNING MATERIAL (SLM)

68 Business Mathematics and Statistics C = [ 2 9 1 6 ] is a row matrix of order 1 × 4.   4   1  D = 9 is a column matrix of order 4 × 1. 6    4.2 Different Types of Matrices (1) Null Matrix: A matrix having all its elements zero is called a null matrix. It may be a rectangular or square matrix. A null matrix is denoted by O.  0 0 0 0  0 0  0 0 0 0 O=   and O=   0 0  0 0 0 0  are null matrices. Triangular matrix: A square matrix A = (aij)n is called an upper triangular matrix, if aij = 0 for all i > j, i.e., if all elements below the main diagonal are zero.

For examples, the matrices FI F –1 3 I –5 A=G J X=G 0 25 J 24 J H 0 6K G H 0 0 2K are upper triangular matrices of order 2 and 3 respectively. A square matrix A = (aij)n × n is called an lower triangular matrix, if aij = 0 for all i < j, i.e.,, if all element above the main diagonal are zero. For example, the matrices FI FI 2 00 0 30 0 GA = 15 0J B=G J G –1 0J 30 HK G J7 2 –4 0 G 2 –1 6J KH 1 7 3 0 are lower triangular matrices of order 2 and 4 respectively. CU IDOL SELF LEARNING MATERIAL (SLM)

Matrices 69 (2) Diagonal Matrix: A matrix having elements, other than the elements in its principal diagonal all zero is called a diagonal matrix.  2 0 0  0 8 0  4 0 For example, A =   and B =   are diagonal matrices.. 0 0 6  0 5  (3) Scalar Matrix: If the elements of a diagonal matrix are all equal, it is known as a scalar matrix.  8 0 0  4 0  0 8 0 For example, A =   and B =   are scalar matrices.  0 4 0 0 8  (4) Unit Matrix: If each of the diagonal elements of a square matrix is equal to 1 (unity) and the other elements are all zero, then such a matrix is called a Unit matrix or Identity matrix. Unit matrix is denoted by the letter I.  1 0 0  1 0

 0 1 0 For example, I=  and I =   0 0 1  0 1  are unit matrices. (5) Equal Matrices: Two matrices of the same order are said to be equal if their corresponding elements are equal. aa bb  11 12   11 12  a aFor example, A =   band B =  b   21 22   21 22  are equal if a11 = b11 , a12 = b12 , a21 = b21 , a22 = b22. CU IDOL SELF LEARNING MATERIAL (SLM)

70 Business Mathematics and Statistics (6) Transpose of a Matrix: If all the rows and columns of a given matrix are interchanged then the resulting matrix is called the transpose of the given matrix. The transpose of a matrix of order p × n is a matrix of order n × p. The transpose of a matrix A is denoted by ‘A’. 3 1 2 4 2 41  6 If, A =   then 15 3 6 3 2 1 1 4  5   its transpose A' =  2 1 4 6 3 6  (7) Coefficient Matrix: Given a set of 3 linear homogenous equations, the matrix formed by the coefficient of these linear equations, is called a coefficient matrix. Given 3x + 2y + 4z = 19 2x – y + z = 3 6x + 2y – z = 17

The coefficient matrix is: 32 4 1  1 2   1 62 (8) Augumented Matrix: With respect to the linear homogenous equations 2x + 3y – z = 9 x+y+z=9 3x – y – z = – 1 CU IDOL SELF LEARNING MATERIAL (SLM)