Page 64 - DMTH201_Basic Mathematics-1

Unit 3: Matrix

If the prices per unit of A, B and C are 2.50, 1.25 and 1.50 and the costs per unit are 1.70, Notes
1.20 and 0.80 respectively, find total profit in each market by using matrix algebra.
Solution:

Let Q be the matrix of the quantities sold.
8000 4000 16000
Q = 7000 18000 9000

2.50 1.70
We can also write P 1.25 and C 1.20 , as the matrices of prices and costs respectively.
1.50 0.80
Note: P and C can also be written as row matrices.
The respective total revenue and cost matrices are
2.50
8000 4000 16000 49000
TR = QP = 1.25 = ,
7000 18000 9000 53500
1.50
1.70
8000 4000 16000 31200
and TC = QC = 1.20 =
7000 18000 9000 40700
0.80
49000 31200 17800
The profit matrix = TR – TC = –
53500 40700 12800
Hence the profits from market I and II are 17,800 and 12,800 respectively.
Alternatively, the profit matrix can be written as = Q [P – C].

Example
A firm produces three products P , P and P requiring the mix-up of three materials M , M and
1 2 3 1 2
M . The per unit requirement of each product for each material (in units) is as follows:
3
M M M
1 2 3
P 2 3 1
1
A = P 4 2 5
2
P 2 4 2
3
Using matrix notations, find:
(i) The total requirement of each material if the firm produces 100 units of each product.

(ii) The per unit cost of production of each product if the per unit cost of materials M , M and
1 2
M are 5, 10 and 5 respectively.
3
(iii) The total cost of production if the firm produces 200 units of each product.
Solution:
100 P 1 5 M 1
Let B 100 P and C 10 M denote the output vector and the cost of material vector
2
2
100 P 5 M
3 3
respectively.

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