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 3, 2 and 1 students respectively with a total award money of Rs 1,600. School B wants to spend Rs 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 amount of award 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

Given, $x+y+z=900$
$3x+2y+z=1600$
$4x+y+3z=2300$
$\text{The given system of linear equation may be written in}$ $\text{matrix form as: AX=B ......(i)}$
$\text{where,}$
$A=\left[\begin{array}{ccc}1& 1& 1\\ 3& 2& 1\\ 4& 1& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}900\\ 1600\\ 2300\end{array}\right]$

$\text{co-factors of elements of first row are}$
$A_{11}=\left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|=6-1=5,$

$A_{12}=-\left|\begin{array}{cc}3& 1\\ 4& 3\end{array}\right|=-(9-4)=-5$

$A_{13}=\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|=3-8=-5$

$\text{co-factors of elements of second row are}$

$A_{21}=-\left|\begin{array}{cc}-1& 1\\ 1& 3\end{array}\right|=-(3-1)=-2$

$A_{22}=-\left|\begin{array}{cc}1& 1\\ 4& 3\end{array}\right|=3-4=-1$

$A_{23}=-\left|\begin{array}{cc}1& 1\\ 4& 1\end{array}\right|=-(1-4)=3$

$\text{co-factors of elements of 3rd row are}$

$A_{31}=\left|\begin{array}{cc}1& 1\\ 2& 1\end{array}\right|=1-2=-1$

$A_{32}=-\left|\begin{array}{cc}1& 1\\ 3& 1\end{array}\right|=-(1-3)=2$

$A_{33}=-\left|\begin{array}{cc}1& 1\\ 3& 2\end{array}\right|=2-3=-1$

$|A|=a_{11}{A}_{11}+a_{12}{A}_{12}+a_{13}{A}_{13}$

$=1\times 5+1\times (-5)+1\times (-5)=-5$

$\text{Adj A}={\left[\begin{array}{ccc}5& -5& -5\\ -2& -1& 3\\ -1& 2& -1\end{array}\right]}^T=\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right]$

$A^{-1}=\frac{1}{|A|}AdjA=\frac{1}{-5}\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right]$

$\text{from (i),}$ $X=A^{-1}B$

$=-\frac{1}{5}\left[\begin{array}{ccc}5& -2& -1\\ -5& -1& 2\\ -5& 3& -1\end{array}\right]\left[\begin{array}{c}900\\ 1600\\ 2300\end{array}\right]$

$=-\frac{1}{5}\left[\begin{array}{c}4500-3200-2300\\ -4500-1600+4600\\ -4500+4800-2300\end{array}\right]$

$=-\frac{1}{5}\left[\begin{array}{c}-1000\\ -1500\\ -2000\end{array}\right]=\left[\begin{array}{c}200\\ 300\\ 400\end{array}\right]$

$\Rightarrow x=200,y=300,z=400$

$\text{Thus, value of each award is Rs 200,Rs 300 and Rs 400.}$
$\text{Punctuality should also be consider for award.}$