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. 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?

Answer 1

\[A . T . Q\]
\[4x + 3y + 2z = 37000\]
\[5x + 3y + 4z = 47000\]
\[x + y + z = 12000\]
We can expressed these equations as AX = B .
\[\text{ Where }A = \begin{bmatrix}4 & 3 & 2 \\ 5 & 3 & 4 \\ 1 & 1 & 1\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}37000 \\ 47000 \\ 12000\end{bmatrix}\]
\[\left| A \right| = 4\left( 3 - 4 \right) - 3\left( 5 - 4 \right) + 2\left( 5 - 3 \right) = - 4 - 3 + 4 = - 3 \neq 0\]
So, A is non singular therefore inverse exists . 
\[ A_{11} = - 1 A_{12} = - 1 A_{13} = 2\]
\[ A_{21} = - 1 A_{22} = 2 A_{23} = - 1\]
\[ A_{31} = 6 A_{32} = - 6 A_{33} = - 3\]
\[adj A = \begin{bmatrix}- 1 & - 1 & 6 \\ - 1 & 2 & - 6 \\ 2 & - 1 & - 3\end{bmatrix}\]
\[ A^{- 1} = \frac{1}{\left| A \right|}adj A = - \frac{1}{3}\begin{bmatrix}- 1 & - 1 & 6 \\ - 1 & 2 & - 6 \\ 2 & - 1 & - 3\end{bmatrix}\]
\[X = A^{- 1} B = - \frac{1}{3}\begin{bmatrix}- 1 & - 1 & 6 \\ - 1 & 2 & - 6 \\ 2 & - 1 & - 3\end{bmatrix} \begin{bmatrix}37000 \\ 47000 \\ 12000\end{bmatrix}\]
\[ = - \frac{1}{3}\begin{bmatrix}- 37000 - 47000 + 72000 \\ - 37000 + 94000 - 72000 \\ 74000 - 47000 - 36000\end{bmatrix} = - \frac{1}{3}\begin{bmatrix}- 12000 \\ - 15000 \\ - 9000\end{bmatrix} = \begin{bmatrix}4000 \\ 5000 \\ 3000\end{bmatrix}\]
\[So, x = 4000 , y = 5000\text{ and }z = 3000 .\]