In a legislative assembly election, a political party hired a public relation firm to promote its candidate
in three ways: telephone, house calls and letters. The numbers of contacts of each type in three cities A,
B & C are (500, 1000, 5000), (3000, 1000, 10000) and (2000,1500,4000), respectively. The party paid
Rs.3700, Rs.7200, and Rs.4300 in cities A, B & C respectively. Find the costs per contact using matrix
method. Keeping in mind the economic condition of the country, which way of promotion is better in
your view?
Answers
In a Legislative assembly election, a political party hired a public relation firm to promote its candidate in three ways: telephone, house calls and letters.
To find : Find the costs per contact using matrix method.
Solution:
500x + 1000y + 5000z = 3700
300x + 1000y + 10000z = 7200
2000x + 1500y + 4000z = 4300
\begin{gathered}\begin{bmatrix} 500 & 1000 & 5000 \\ 3000 & 1000 & 10000 \\ 2000 & 1500 & 4000 \end{bmatrix}\end{gathered}
⎣
⎢
⎡
500
3000
2000
1000
1000
1500
5000
10000
4000
⎦
⎥
⎤
\begin{gathered}\begin{bmatrix} x \\ y \\ z \end{bmatrix}\end{gathered}
⎣
⎢
⎡
x
y
z
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⎤
= \begin{gathered}\begin{bmatrix} 3700 \\ 7200 \\ 4300 \end{bmatrix}\end{gathered}
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⎡
3700
7200
4300
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⎤
=> 100 \begin{gathered}\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}\end{gathered}
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⎡
5
30
20
10
10
15
50
100
40
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⎥
⎤
\begin{gathered}\begin{bmatrix} x \\ y \\ z \end{bmatrix}\end{gathered}
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⎢
⎡
x
y
z
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⎤
= 100 \begin{gathered}\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}\end{gathered}
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⎡
37
72
43
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⎤
=> \begin{gathered}\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}\end{gathered}
⎣
⎢
⎡
5
30
20
10
10
15
50
100
40
⎦
⎥
⎤
\begin{gathered}\begin{bmatrix} x \\ y \\ z \end{bmatrix}\end{gathered}
⎣
⎢
⎡
x
y
z
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⎥
⎤
= \begin{gathered}\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}\end{gathered}
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⎡
37
72
43
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AX = B
=> X = A⁻¹B
A = \begin{gathered}\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}\end{gathered}
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⎢
⎡
5
30
20
10
10
15
50
100
40
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⎤
A⁻¹ = adjA / | A |
|A| = 5(-1100) - 10(-800) + 50(250)
= -5500 + 8000 + 12500
= 15000
adj A
A11 = -1100
A12 = 800
A13 = 250
A21 = 350
A22 = -800
A23 = 125
A31 = 500
A32 = 1000
A33 = -250
adjA = \begin{gathered}\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}\end{gathered}
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−1100
800
250
350
−800
125
500
1000
−250
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A⁻¹ = (1/15000) \begin{gathered}\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}\end{gathered}
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−1100
800
250
350
−800
125
500
1000
−250
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⎤
X = A⁻¹B
= (-1/1000) \begin{gathered}\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}\end{gathered}
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−1100
800
250
350
−800
125
500
1000
−250
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⎥
⎤
\begin{gathered}\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}\end{gathered}
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⎡
37
72
43
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= (1/15000) \begin{gathered}\begin{bmatrix} 6000 \\ 15000 \\ 7500 \end{bmatrix}\end{gathered}
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⎡
6000
15000
7500
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= \begin{gathered}\begin{bmatrix} 0.4 \\ 1 \\ 0.5 \end{bmatrix}\end{gathered}
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0.4
1
0.5
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⎤
x = 0.4
y = 1
z = 0.5
costs per contact
telephone, = 0.4
house calls =1
and letters = 0.5