Two institutions decided to award their employees for the three values resourcefulness, competence and determination in the from of prizes at the rate of₹x, ₹y and ₹z respectively per person. The first institutions decided to award respectively 4,3 and 2 employees with total prize money of ₹37000 and the second institution decided to award respectively 5,3 and 4 employees with total prize money of ₹47000. If all the three prizes per person together amount to ₹12000, then using matrix method find the value of x,y and z. What values are described in this equations?
Let the numbers are x, y, z be the cash awards for Resourcefulness, Competence, and Determination respectively
4x + 3y + 2z = 37000 …… (i)
Also,
5x + 3y + 4z = 47000 …… (ii)
Again,
x + y + z = 12000 …… (iii)
A X = B
|A| = 4(3 – 4) – 3(5 – 4) + 2(5 – 3)
= 4(– 1) – 3(1) + 2(2)
= – 4 – 3 + 4
= – 3
Hence, the unique solution given by x = A – 1B
C11 = (– 1)1 + 1 (3 – 4) = – 1
C12 = (– 1)1 + 2 (5 – 4) = – 1
C13 = (– 1)1 + 3 (5 – 3) = 2
C21 = (– 1)2 + 1 (3 – 2) = – 1
C22 = (– 1)2 + 2 (4 – 2) = 2
C23 = (– 1)2 + 3 (4 – 3 ) = – 1
C31 = (– 1)3 + 1 (12 – 6) = 6
C32 = (– 1)3 + 2 (16 – 10) = – 6
C33 = (– 1)3 + 3 (12 – 15) = – 3
Adj A =
X = A – 1 B =
X =
X =
=
Hence, x = 4000, y = 5000 and z = 3000
Thus, The value X, Y, Z describes the amount of prizes per person for Resourcefulness, Competence and Determination.