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Math 120 SP08
Notes & Examples for Section 3.5: InputOutput Models
Brodnick
Application of Matrices:
Used to analyze economy by considering how sectors interrelate.
Consider 2 sectors, Sector 1: Oil Sector, and Sector 2: Electricity Sector
Both sectors produce a product and we can measure the products by their dollar value.
(Let 1 unit = $1 worth of that product)
Consider the following scenario:
To produce one unit of oil, the oil sector uses 30¢ worth of oil (to power machinery) and 20¢ of electricity.
To produce one unit of electricity, the electricity sector uses 25¢ worth of electricity and 10¢ worth of oil.
10,000 units of oil/year
18,000 units of electricity/year
Question
: How many units should each sector provide to meet both internal and external demands?
i.e., Total Supply = Total Demand
Unknown:
1
x
= total supply units from oil
2
x
= total supply units from electricity
So
10000
10
.
0
30
.
0
2
1
1
+
+
=
x
x
x
18000
25
.
0
20
.
0
2
1
2
+
+
=
x
x
x
System of two linear equations in two unknowns:
18000
25
.
0
2
.
0
10000
1
.
0
3
.
0
2
1
2
2
1
1
+
+
=
+
+
=
x
x
x
x
x
x
Rewrite system in matrix form:
+
=
18000
10000
25
.
0
2
.
0
1
.
0
3
.
0
2
1
2
1
x
x
x
x
In symbols:
D
AX
X
+
=
In our example, using this formula would give the following:
y
electricit
of
units
oil
of
units
9
.
28910
8
.
18415
18000
10000
18000
10000
75
.
0
2
.
0
1
.
0
7
.
0
18000
10000
25
.
0
2
.
0
1
.
0
3
.
0
1
0
0
1
101
140
101
40
101
20
101
150
1
1
2
1
→
→
≈
⋅
=
⋅


=
⋅

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 Spring '08
 Brodnick
 Math, Matrices

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