Question

Two schools $P$ and $Q$ want to award their selected students on the values of Discipline, Politeness, and Punctually. The school $P$ wants to award $Rs.x$ each, $y$ 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.

Answer 1

Given:
$3x+2y+z=1000$
$4x+y+3z=1500$
$x+y+z=600$
The corresponding system of matrix equation is:
$AX=B$ where $A=\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\&B=\left[\begin{array}{c}1000\\ 1500\\ 600\end{array}\right]$
$|A|=\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right]=3(1-3)-2(4-3)+1(4-1)=-6-2+3=-5$
$A^{-1}=\frac{1}{|A|}adjA=\frac{1}{-5}\left[\begin{array}{ccc}-2& -1& 5\\ -1& 2& -5\\ 3& -1& -5\end{array}\right]$
$X=A^{-1}B=\left[\begin{array}{c}100\\ 200\\ 300\end{array}\right]$
Therefore, $X=Rs.100,Y=Rs.200,Z=Rs.300.$

Apart from these values there can be value of being friendly. responsible.