Question

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs. $6,000$. Three times the award money for Hard work added to that given for honesty amounts to Rs. $11,000$. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Answer 1

Let Honesty award $=x$, Regularity award $=y$, Hard work award $=z$

According to the question, $x+y+z=6000$
$3z+x=110000$
$x+z=2y$
$x+y+z=6000$

$x+0y+3z=11000$
$x-2y+z=0$

We can represent these equations using Matrices as:
$\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 3\\ 1& -2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6000\\ 11000\\ 0\end{array}\right]$

$AX=B$

Therefore, $X=A^{-1}B$

Now $\left|A\right|=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 3\\ 1& -2& 1\end{array}\right]=6+2-2=6\ne 0$

Therefore, $A^{-1}$ exists.

$Adj.A=\left[\begin{array}{ccc}6& -3& 3\\ 2& 0& -2\\ -2& -3& -1\end{array}\right]$

$A^{-1}=\frac{1}{\left|A\right|}(Adj.A)$

= $\frac{1}{6}\left[\begin{array}{ccc}6& -3& 3\\ 2& 0& -2\\ -2& -3& -1\end{array}\right]$

$X=A^{-1}B=\frac{1}{6}\left[\begin{array}{ccc}6& -3& 3\\ 2& 0& -2\\ -2& -3& -1\end{array}\right]\left[\begin{array}{c}6000\\ 11000\\ 0\end{array}\right]$

$=\frac{1}{6}\left[\begin{array}{c}36000-33000+0\\ 12000+0-0\\ -12000+33000-0\end{array}\right]$

$=\frac{1}{6}\left[\begin{array}{c}3000\\ 12000\\ 21000\end{array}\right]$

$=\left[\begin{array}{c}500\\ 2000\\ 35000\end{array}\right]$

Therefore, Award for honesty = Rs. $500$

Award for regularity = Rs. $2000$

Award for hard work = Rs. $3500$

One more value can be punctuality.