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 ₹1,000. 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.

Answer 1

​Let the award money given for Discipline, Politeness and Punctuality be ₹x, ₹y and ₹z respectively.

Since, the total cash award is ₹600.
∴ x + y + z = 600 ....(1)

Award money given by school P is ₹1,000.
∴ 3x + 2y + z = 1000 ....(2)

Award money given by school Q is ₹1,500.
∴ 4x + y + 3z = 1500 ....(3)

The above system of equations can be written in matrix form AX = B as
$$\begin{gathered} \left[\begin{array}{ccc}1& 1& 1\\ 3& 2& 1\\ 4& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}600\\ 1000\\ 1500\end{array}\right] \\ \mathrm{Where},A=\left[\begin{array}{ccc}1& 1& 1\\ 3& 2& 1\\ 4& 1& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mathrm{and}B=\left[\begin{array}{c}600\\ 1000\\ 1500\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{ccc}1& 1& 1\\ 3& 2& 1\\ 4& 1& 3\end{array}\right| \\ =1\left(6-1\right)-1\left(9-4\right)+1(3-8) \\ =5-5-5 \\ =-5 \\ \mathrm{Let}{C}_{ij}\mathrm{be}\mathrm{the}\mathrm{cofactors}\mathrm{of}\mathrm{elements}{a}_{ij}\mathrm{in}A=\left[a_{ij}\right].\mathrm{Then}, \\ {C}_{11}={\left(-1\right)}^{1+1}\left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|=5,C_{12}={\left(-1\right)}^{1+2}\left|\begin{array}{cc}3& 1\\ 4& 3\end{array}\right|=-5,C_{13}={\left(-1\right)}^{1+3}\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|=-5 \\ {C}_{21}={\left(-1\right)}^{2+1}\left|\begin{array}{cc}1& 1\\ 1& 3\end{array}\right|=-2,C_{22}={\left(-1\right)}^{2+2}\left|\begin{array}{cc}1& 1\\ 4& 3\end{array}\right|=-1,C_{23}={\left(-1\right)}^{2+3}\left|\begin{array}{cc}1& 1\\ 4& 1\end{array}\right|=3 \\ {C}_{31}={\left(-1\right)}^{3+1}\left|\begin{array}{cc}1& 1\\ 2& 1\end{array}\right|=-1,C_{32}={\left(-1\right)}^{3+2}\left|\begin{array}{cc}1& 1\\ 3& 1\end{array}\right|=2,C_{33}={\left(-1\right)}^{3+3}\left|\begin{array}{cc}1& 1\\ 3& 2\end{array}\right|=-1 \\ \mathrm{adj}A={\left[\begin{array}{ccc}5& -5& -5\\ -2& -1& 3\\ -1& 2& -1\end{array}\right]}^T \\ =\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right] \\ \Rightarrow A^{-1}=\frac{1}{\left|A\right|}\mathrm{adj}A \\ =\frac{1}{-5}\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right] \\ X=A^{-1}B \\ \Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{ccc}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}{c}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}{c}x\\ y\\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{c}-500\\ -1000\\ -1500\end{array}\right] \\ \Rightarrow x=\frac{-500}{-5},y=\frac{-1000}{-5}\mathrm{and}z=\frac{-1500}{-5} \\ \therefore x=100,y=200\mathrm{and}z=300. \end{gathered}$$

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.