Two schools A and B want to award their selected students on the values of sincerity,

Solution:

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

Since, the total cash award is ₹ 900 .

$\therefore x+y+z=900$          ....(1)

Award money given by school $A$ is ₹ 1,600 .

$\therefore 3 x+2 y+z=1600$      ....(2)

Award money given by school $B$ is $₹ 2,300$.

$\therefore 4 x+y+3 z=2300$               ....(3)

The above system of equations can be written in matrix form CX = D as

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

Where, $C=\left[\begin{array}{lll}1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $D=\left[\begin{array}{c}900 \\ 1600 \\ 2300\end{array}\right]$

Now,

$|C|=\left|\begin{array}{lll}1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3\end{array}\right|$

$=1(6-1)-1(9-4)+1(3-8)$

$=5-5-5$

$=-5$

Let $C_{i j}$ be the cofactors of elements $c_{i j}$ in $C=\left[c_{i j}\right]$. Then,

$C_{11}=(-1)^{1+1}\left|\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right|=5, \quad C_{12}=(-1)^{1+2}\left|\begin{array}{ll}3 & 1 \\ 4 & 3\end{array}\right|=-5, \quad C_{13}=(-1)^{1+3}\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|=-5$

$C_{21}=(-1)^{2+1}\left|\begin{array}{ll}1 & 1 \\ 1 & 3\end{array}\right|=-2, \quad C_{22}=(-1)^{2+2}\left|\begin{array}{ll}1 & 1 \\ 4 & 3\end{array}\right|=-1, \quad C_{23}=(-1)^{2+3}\left|\begin{array}{ll}1 & 1 \\ 4 & 1\end{array}\right|=3$

$C_{31}=(-1)^{3+1}\left|\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right|=-1, \quad C_{32}=(-1)^{3+2}\left|\begin{array}{ll}1 & 1 \\ 3 & 1\end{array}\right|=2, \quad C_{33}=(-1)^{3+3}\left|\begin{array}{ll}1 & 1 \\ 3 & 2\end{array}\right|=-1$

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

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

$\Rightarrow C^{-1}=\frac{1}{|C|}$ adj $C$

$X=C^{-1} D$

$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{rrr}5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1\end{array}\right]\left[\begin{array}{c}900 \\ 1600 \\ 2300\end{array}\right]$

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

$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=-\frac{1}{5}\left[\begin{array}{l}-1000 \\ -1500 \\ -2000\end{array}\right]$

$\Rightarrow x=\frac{-1000}{-5}, y=\frac{-1500}{-5}$ and $z=\frac{-2000}{-5}$

$\therefore x=200, y=300$ and $z=400$

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.