# Two schools $P$ and $Q$ want to award their selected students on the values of Tolerance,Kindness and Leadership. The school $P$ wants to award $Rs.x,\:Rs.y\:and\:\:Rs.z$ each for the three respective values to $3,2\:and\:1$ students respectively with a total award money of $Rs.2,200.$ School $Q$ wants to spend $Rs.3,100$ to award its 4,1 and 3 students on the respective values (by giving the same award money to the three values as school $P$). If the total amount of award for one prize on each value is $Rs.1,200,$ using matrices, find the award money for each value.

3 equations are formed from the given statements.
$i.e.,$ Given: $3x+2y+z=2200$
$\qquad\:4x+y+3z=3100$ and
$\qquad\:x+y+z=1200$
Converting the the system of equations in matrix form we get
$\begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} 2200 \\ 3100 \\ 1200 \end{bmatrix}$
$i.e.,AX=B$
where $A=\begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix}$
$X= \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ and $B= \begin{bmatrix} 2200 \\ 3100 \\ 1200 \end{bmatrix}$
$\Rightarrow\:X=A^{-1}.B$
$A^{-1}=\large\frac{1}{|A|}$$(Adj\:A) |A|=3(1-3)-2(4-3)+1(4-1)=-6-2+3=-5 Adj\:A=\begin{bmatrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{bmatrix} \therefore\:A^{-1}=\large\frac{1}{-5}$$\begin{bmatrix} -2 & -1 & 5 \\ -1 & 2 & -5 \\ 3 & -1 & -5 \end{bmatrix}$
$=\large\frac{1}{5}$$\begin{bmatrix} 2 & 1 & -5 \\ 1 &- 2 & 5 \\ -3 & 1 & 5 \end{bmatrix} \Rightarrow\:X=\large\frac{1}{5}$$\begin{bmatrix} 2 & 1 & -5 \\ 1 &- 2 & 5 \\ -3 & 1 & 5 \end{bmatrix}\begin{bmatrix} 2200 \\ 3100 \\ 1200 \end{bmatrix}$
$\qquad=\large\frac{1}{5}$$\begin{bmatrix} 4400+3100-6000 \\ 2200-6200+6000 \\ -6600+3100+6000 \end{bmatrix}$
$\qquad=\begin{bmatrix} 300 \\ 400\\ 500 \end{bmatrix}$
$\Rightarrow\:x=300,\:y=400\:\:and\:\:z=500$
$i.e.,$ The award money for each value are
$Rs.300$ for Tolerance, $Rs.400$ for Kindness and $Rs.500$ for Kindness.