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. x, y 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 x, y and z. What values are described in this equations?
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 $z=3000$.