Question

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹1,600. School B wants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Answer 1

Let the award money given for sincerity, truthfulness and helpfulness be ₹x, ₹y and ₹z respectively.

Since, the total cash award is ₹900.
∴ x + y + z = 900 ....(1)

Award money given by school A is ₹1,600.
∴ 3x + 2y + z = 1600 ....(2)

Award money given by school B is ₹2,300.
∴ 4x + y + 3z = 2300 ....(3)

The above system of equations can be written in matrix form CX = D 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}900\\ 1600\\ 2300\end{array}\right] \\ \mathrm{Where},C=\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}D=\left[\begin{array}{c}900\\ 1600\\ 2300\end{array}\right] \\ \mathrm{Now}, \\ \left|C\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}{c}_{ij}\mathrm{in}C=\left[c_{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}C={\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 C^{-1}=\frac{1}{\left|C\right|}\mathrm{adj}C \\ =\frac{1}{-5}\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right] \\ X=C^{-1}D \\ \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}900\\ 1600\\ 2300\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{c}4500-3200-2300\\ -4500-1600+4600\\ -4500+4800-2300\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{c}-1000\\ -1500\\ -2000\end{array}\right] \\ \Rightarrow x=\frac{-1000}{-5},y=\frac{-1500}{-5}\mathrm{and}z=\frac{-2000}{-5} \\ \therefore x=200,y=300\mathrm{and}z=400. \end{gathered}$$

Hence, the award money for each value of sincerity, truthfulness and helpfulness is ₹200, ₹300 and ₹400.

One more value which should be considered for award hardwork.