Two schools P and Q want to award their selected students on the values of Discipline,

Question:

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 ₹1,000School 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.

Solution:

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.

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