Q12 of 75 Page 8

The prices of three commodities P,Q and R and ₹ x,y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. C purchases 1 unit of Q. B purchases of Q and 6 units of R. In the process A, B and C earn ₹6000, ₹5000 and ₹13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using the matrix method.

Let the numbers are x, y, z


3x + 5y – 4 z = 6000 …… (i)


Also,


2x – 3y + z = 5000 …… (ii)


Again,


– x + 4y + 6z = 13000 ……(iii)



A X = B


|A| = 3(– 18 – 4) – 2(30 + 16) – 1(5 – 12)


= 3(– 22) – 2(46) + 7


= – 66 – 92 + 7


= – 151


Hence, the unique solution given by x = A – 1B


C11 = (– 1)1 + 1 (– 18 – 4) = – 22


C12 = (– 1)1 + 2 (12 + 1) = – 13


C13 = (– 1)1 + 3 (8 – 3) = 5


C21 = (– 1)2 + 1 (30 + 16) = – 46


C22 = (– 1)2 + 2 (18 – 4) = 14


C23 = (– 1)2 + 3 (12 + 5 ) = – 17


C31 = (– 1)3 + 1 (5 – 12) = – 7


C32 = (– 1)3 + 2 (3 + 8) = – 11


C33 = (– 1)3 + 3 (– 9 – 10) = – 19


Adj A =


X = A – 1 B =


X =


X =


=


Hence, x = 3000, y = 1000 and z = 2000


More from this chapter

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10

An amount of ₹10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined incomes are ₹1310 and the combined income of first and second investment is ₹ 190 short of the income from the third. Find the investment in each using matrix method.

 11

A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of the third product exceeds the production of a first product by 8 tons while the total production of a first and third product is twice the production of the second product. Determine the production level of each product using the matrix method.

 13

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. A part from these values, namely, honesty cooperation and supervision, suggest one more value which the management must include for awards.

 14

A school wants to award its student for the values of Honesty, Regularity and Hard work with a total cash award of ₹6000. Three times the award money for Hard work added to that given for honesty amounts to ₹11000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award for each value, using the matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, and suggest one more value which the school must include for awards.