Two institutions decided to award their employees for the three values of resourcefulness,

Question:

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. xy and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of xy and z. What values are described in this equations?

Solution:

A.T.Q

$4 x+3 y+2 z=37000$

$5 x+3 y+4 z=47000$

$\mathrm{x}+\mathrm{y}+\mathrm{z}=12000$

We can expressed these equations as $\mathrm{AX}=\mathrm{B}$.

Where $\mathrm{A}=\left[\begin{array}{lll}4 & 3 & 2 \\ 5 & 3 & 4 \\ 1 & 1 & 1\end{array}\right], \mathrm{X}=\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}37000 \\ 47000 \\ 12000\end{array}\right]$

$|\mathrm{A}|=4(3-4)-3(5-4)+2(5-3)=-4-3+4=-3 \neq 0$

So, $A$ is non singular therefore inverse exists.

$\mathrm{A}_{11}=-1 \quad \mathrm{~A}_{12}=-1 \quad \mathrm{~A}_{13}=2$

$\mathrm{A}_{21}=-1 \quad \mathrm{~A}_{22}=2 \quad \mathrm{~A}_{23}=-1$

$\mathrm{A}_{31}=6 \mathrm{~A}_{32}=-6 \mathrm{~A}_{33}=-3$

$\operatorname{adj} A=\left[\begin{array}{ccc}-1 & -1 & 6 \\ -1 & 2 & -6 \\ 2 & -1 & -3\end{array}\right]$

$\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}$ adj $\mathrm{A}=-\frac{1}{3}\left[\begin{array}{ccc}-1 & -1 & 6 \\ -1 & 2 & -6 \\ 2 & -1 & -3\end{array}\right]$

$\mathrm{X}=\mathrm{A}^{-1} \mathrm{~B}=-\frac{1}{3}\left[\begin{array}{ccc}-1 & -1 & 6 \\ -1 & 2 & -6 \\ 2 & -1 & -3\end{array}\right]\left[\begin{array}{l}37000 \\ 47000 \\ 12000\end{array}\right]$

$=-\frac{1}{3}\left[\begin{array}{c}-37000-47000+72000 \\ -37000+94000-72000 \\ 74000-47000-36000\end{array}\right]=-\frac{1}{3}\left[\begin{array}{c}-12000 \\ -15000 \\ -9000\end{array}\right]=\left[\begin{array}{l}4000 \\ 5000 \\ 3000\end{array}\right]$

So, $x=4000, y=5000$ and $z=3000$.

 

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