Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
Let the award money given for Discipline, Politeness and Punctuality be $₹ x, ₹ y$ and $₹ z$ respectively.
Since, the total cash award is ₹ 600 .
$\therefore x+y+z=600$ ....(1)
Award money given by school $P$ is ₹ 1,000 .
$\therefore 3 x+2 y+z=1000$ ....(2)
Award money given by school $Q$ is $₹ 1,500$.
$\therefore 4 x+y+3 z=1500$ .....(3)
The above system of equations can be written in matrix form AX = B as
$\left[\begin{array}{lll}1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}600 \\ 1000 \\ 1500\end{array}\right]$
Where, $A=\left[\begin{array}{lll}1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $B=\left[\begin{array}{c}600 \\ 1000 \\ 1500\end{array}\right]$
Now,
$|A|=\left|\begin{array}{lll}1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3\end{array}\right|$
$=1(6-1)-1(9-4)+1(3-8)$
$=5-5-5$
$=-5$
Let $C_{i j}$ be the cofactors of elements $a_{i j}$ in $A=\left[a_{i j}\right] .$ Then,
$C_{11}=(-1)^{1+1}\left|\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right|=5, \quad C_{12}=(-1)^{1+2}\left|\begin{array}{ll}3 & 1 \\ 4 & 3\end{array}\right|=-5, \quad C_{13}=(-1)^{1+3}\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|=-5$
$C_{21}=(-1)^{2+1}\left|\begin{array}{ll}1 & 1 \\ 1 & 3\end{array}\right|=-2, \quad C_{22}=(-1)^{2+2}\left|\begin{array}{ll}1 & 1 \\ 4 & 3\end{array}\right|=-1, \quad C_{23}=(-1)^{2+3}\left|\begin{array}{ll}1 & 1 \\ 4 & 1\end{array}\right|=3$
$C_{31}=(-1)^{3+1}\left|\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right|=-1, \quad C_{32}=(-1)^{3+2}\left|\begin{array}{ll}1 & 1 \\ 3 & 1\end{array}\right|=2, \quad C_{33}=(-1)^{3+3}\left|\begin{array}{ll}1 & 1 \\ 3 & 2\end{array}\right|=-1$
adj $A=\left[\begin{array}{rrr}5 & -5 & -5 \\ -2 & -1 & 3 \\ -1 & 2 & -1\end{array}\right]^{T}$
$=\left[\begin{array}{rrr}5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1\end{array}\right]$
$\Rightarrow A^{-1}=\frac{1}{|A|}$ adj $A$
$=\frac{1}{-5}\left[\begin{array}{rrr}5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1\end{array}\right]$
$X=A^{-1} B$
$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{rrr}5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1\end{array}\right]\left[\begin{array}{c}600 \\ 1000 \\ 1500\end{array}\right]$
$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{c}3000-2000-1500 \\ -3000-1000+3000 \\ -3000+3000-1500\end{array}\right]$
$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{l}-500 \\ -1000 \\ -1500\end{array}\right]$
$\Rightarrow x=\frac{-500}{-5}, y=\frac{-1000}{-5}$ and $z=\frac{-1500}{-5}$
$\therefore x=100, y=200$ and $z=300$
Hence, the award money for each value of Discipline, Politeness and Punctuality is ₹ 100 , ₹ 200 and ₹ 300 .
One more value which should be considered for award is Honesty.