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 Rs.x each, Rs.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.1600. School B wants to spend Rs.2300 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.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 spent on the values of sincerity, truthfifiness and helpfulness he Rs.x each. Rs.y each and Rs.z each respectively.

$\therefore 3x+2y+z=1600,4x+y+3z=2300,x+y+z=900$

The given situation can be expressed as: $\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1600\\ 2300\\ 900\end{array}\right]$

where $A=\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right],x=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}1600\\ 2300\\ 900\end{array}\right]\Rightarrow AX=BX=A^{-1}B\dots \left(i\right)$

Now, $|A|=3(1-3)-2(4-3)+1(4-1)=-5\ne 0,$ so $A^{-1}$ exists.

Consider $A_{ij}$ as the cofactors of the element $a_{ij}$ of matrix A.

$A_{11}=-2,A_{12}=-1,A_{13}==3A_{21}=-1,A_{22}=2,A_{23}=-1A_{31}=5,A_{32}=-5,A_{33}=-5$

So, $adj.A=\left[\begin{array}{ccc}-2& -1& 5\\ -1& 2& -5\\ 3& -1& -5\end{array}\right]\therefore A^{-1}=\frac{1}{|A|}adj.A=\frac{1}{5}\left[\begin{array}{ccc}2& 1-5& \\ 1& -2& 5\\ -3& 1& 5\end{array}\right]$

By (i), $X=\frac{1}{5}\frac{1}{|A|}adj.A=\frac{1}{5}\left[\begin{array}{ccc}2& 1-5& \\ 1& -2& 5\\ -3& 1& 5\end{array}\right]\left[\begin{array}{c}1600\\ 2300\\ 900\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\left[\begin{array}{c}200\\ 300\\ 400\end{array}\right]$

By equality of matrices, we get : x = 200, y = 300, z = 400 .

Hence the award money for the values of sincerity, tnithfulness and helpfulness is Rs.200, Rs.300 and Rs.400 respectively.

Also, the value of Obedience can be included for the awards.