Two schools P and Q want to award their selected students on the values of Discipline, Politeness, and Punctuality. School P wants to award Rs.x each, Rs.y each and Rs.z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs.1,000. School Q wants to spend Rs.1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs.600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.<p></p>

Two schools P and Q want to award their selected students on the values of Discipline, Politeness, and Punctuality. School P wants to award Rs.x each, Rs.y each and Rs.z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs.1,000. School Q wants to spend Rs.1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs.600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Solution:

Given:
3x+2y+z=1000
4x+y+3z=1500
x+y+z=600
The corresponding system of matrix equation is:
AX=B
where
A=\begin{bmatrix} 3 & 2 & 1 \ 4 & 1 & 3 \ 1 & 1 & 1 \end{bmatrix}, X=\begin{bmatrix} x \ y \ z \end{bmatrix} B=\begin{bmatrix} 1000 \ 1500 \ 600 \end{bmatrix}
|A|=\begin{vmatrix} 3 & 2 & 1 \ 4 & 1 & 3 \ 1 & 1 & 1 \end{vmatrix}=3(1-3)-2(4-3)+1(4-1)=-6-2+3=-5
A^{-1}=\frac{1}{|A|}adjA=\frac{1}{-5}\begin{bmatrix} -2 & 1 & 5 \ 1 & 2 & -5 \ 3 & -1 & -5 \end{bmatrix}
X=A^{-1}B=\begin{bmatrix} 100 \ 200 \ 300 \end{bmatrix}
Therefore, x=Rs.100, y=Rs.200, z=Rs.300.
Apart from these values there can be a value of being friendly and responsible.

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