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 total award money of Rs. $1600$. 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 total 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

Let the numbers are $x,y,z$ be the prize amount per person for sincerity, truthfulness and helpfulness respectively

As per the given data we get,

$3x+2y+z=1600$

$4x+y+3z=2300$

$x+y+z=900$

These three equations can be written as

$\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1600\\ 2300\\ 900\end{array}\right]$

$AX=B$

$\left|A\right|=3(1–3)–2(4–3)+1(4–1)$

$=-6-2+3=-5$

Hence, the unique solution given by $X=A^{-1}B$

$C_{11}={(-1)}^{1+1}\left|\begin{array}{cc}1& 3\\ 1& 1\end{array}\right|=1-3=-2$

$C_{12}={(-1)}^{1+2}\left|\begin{array}{cc}4& 3\\ 1& 1\end{array}\right|=-(4-3)=-1$

$C_{13}={(-1)}^{1+3}\left|\begin{array}{cc}4& 1\\ 1& 1\end{array}\right|=4-1=3$

$C_{21}={(-1)}^{2+1}\left|\begin{array}{cc}2& 1\\ 1& 1\end{array}\right|=-(2-1)=-1$

$C_{22}={(-1)}^{2+2}\left|\begin{array}{cc}3& 1\\ 1& 1\end{array}\right|=3-1=2$

$C_{23}={(-1)}^{2+3}\left|\begin{array}{cc}3& 2\\ 1& 1\end{array}\right|=-(3-2)=-1$

$C_{31}={(-1)}^{3+1}\left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|=6-1=5$

$C_{32}={(-1)}^{3+2}\left|\begin{array}{cc}3& 1\\ 4& 3\end{array}\right|=-(9-4)=-5$

$C_{33}={(-1)}^{3+3}\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|=3-8=-5$

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

$=\left[\begin{array}{ccc}-2& -1& 5\\ -1& 2& -5\\ 3& -1& -5\end{array}\right]$

$X=A^{-1}B=\frac{1}{\left|A\right|}\left(adj\left(A\right)\right)B$

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

$=\left[\begin{array}{c}640+460-900\\ -320-920+900\\ -960+460+900\end{array}\right]$

$=\left[\begin{array}{c}200\\ 300\\ 400\end{array}\right]$

Hence $x=200,y=300$ and $z=400$

VALUE:Excellence in extra curricular activities should be another value considered for an award.