Question

I 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 `3700, `7200, and `4300 in cities A, B & C respectively. Find the costs per contact using matrix method.

Answer 1

Given :  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

$\begin{bmatrix} 500 & 1000 & 5000 \\ 3000 & 1000 & 10000 \\ 2000 & 1500 & 4000 \end{bmatrix}$ $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$  =  $\begin{bmatrix} 3700 \\ 7200 \\ 4300 \end{bmatrix}$

=> 100  $\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}$ $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$  = 100 $\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}$

=>  $\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}$ $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$  =  $\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}$

AX = B

=> X = A⁻¹B

A =  $\begin{bmatrix} 5 & 10 & 50 \\ 30 & 10 & 100 \\ 20 & 15 & 40 \end{bmatrix}$

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  =   $\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}$

A⁻¹ =  (1/15000)   $\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}$

X = A⁻¹B

=  (-1/1000)   $\begin{bmatrix} -1100 & 350 & 500 \\ 800 & -800 & 1000 \\ 250 & 125 & -250 \end{bmatrix}$ $\begin{bmatrix} 37 \\ 72 \\ 43 \end{bmatrix}$

=  (1/15000) $\begin{bmatrix} 6000 \\ 15000 \\ 7500 \end{bmatrix}$

=  $\begin{bmatrix} 0.4 \\ 1 \\ 0.5 \end{bmatrix}$

x = 0.4

y = 1

z = 0.5

costs per contact  

telephone,  = 0.4

house calls  =1

and letters = 0.5

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