Page 108 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

Notes
x 1 1 2 5 2 85 1 135 15
and x = 2 4 7 105 180 20
2 9 9
x 5 1 4 110 90 10
3
Hence x = 15, x = and x = 10.
1 2 3
(ii) The number of vehicles of each type, that can be transported, are given by

1 2 3 10 140 V
1
3 2 2 20 = 130 V 2
2 3 2 30 140 V
3

4.10 Economic Applications

Two-Commodity Market Equilibrium

Let the demand and supply equations of the two commodities, 1 and 2, be as given below:
d
Q = a + b P + c P
1 1 1 1 1 2
s
Q = e + f P + g P
1 1 1 1 1 2
d
Q = a + b P + c P
2 2 2 1 2 2
s
Q = e + f P + g P
2 2 2 1 2 2
s
The two-commodity market will be in equilibrium if Q d 1 Q and Q 2 d Q 2 s .
1
The first condition implies that
a + b P + c P = e + f P + g P
1 1 1 1 2 1 1 1 1 2
or (b – f )P +(c – g )P = –(a – e ) ...(1)
1 1 1 1 1 2 1 1
Similarly, the second equilibrium condition implies that
a + b P + c P = e + f P + g P
2 2 1 2 2 2 2 1 2 2
or (b – f )P + (c – g )P = –(a – e ) ... (2)
2 2 1 2 2 2 2 2
Let us assume, for convenience, that
a = b – f , b = c – g , and g = –(a – e )(i = 1, 2)
i i i i i i i i
Thus, equations (1) and (2) can be written as

P P P
1 1 1 2 1 1 1 1 1
or
P P P
2 1 2 2 2 2 2 2 2
This is a system of two equations in two unknowns P and P , which can be solved either by
1 2
matrix inversion method or by Cramer’s Rule. We shall, however, solve this by matrix inversion
method.

1 1
Let A = , |A| = a b – a b
1 2
2 1
2 2

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