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

₹ x

each,

₹ y

each and

₹ z

each for the three respective values to

3, 2

and

1

students respectively with a total award money of

₹ 1, 600

. School

B

wants to spend

₹ 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

₹ 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.

Accepted Answer

Given that 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 $R s x$ each, $R s y$ each and $R s z$ each for the three respective values to $3, 2$ and $1$ students respectively with a total award money of $R s 1, 600$ . School $B$ wants to spend $R s 2, 300$ 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 $R s 900$ , using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.
Let the award money given forsincerity, truthfulness and helpfulness be $R s x, R s y$ and $R s z$ respectively.

Since, the total cash award is $R s 900$ .

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

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

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

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

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

The above system of equations can be written in matrix form $C X = D$ as

$[\begin{matrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 900 \\ 1600 \\ 2300 \end{matrix}]$

Where, $C = [\begin{matrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}]$ and $D = [\begin{matrix} 900 \\ 1600 \\ 2300 \end{matrix}]$

Now,

$|C| = |\begin{matrix} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{matrix}|$

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

$= 5 - 5 - 5$

$= - 5$

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

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

$C_{21} = {(- 1)}^{2 + 1} |\begin{matrix} 1 & 1 \\ 1 & 3 \end{matrix}| = - 2, C_{22} = {(- 1)}^{2 + 2} |\begin{matrix} 1 & 1 \\ 4 & 3 \end{matrix}| = - 1, C_{23} = {(- 1)}^{2 + 3} |\begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}| = 3$

$C_{31} = {(- 1)}^{3 + 1} |\begin{matrix} 1 & 1 \\ 2 & 1 \end{matrix}| = - 1, C_{32} = {(- 1)}^{3 + 2} |\begin{matrix} 1 & 1 \\ 3 & 1 \end{matrix}| = 2, C_{33} = {(- 1)}^{3 + 3} |\begin{matrix} 1 & 1 \\ 3 & 2 \end{matrix}| = - 1$

$a d j C = {[\begin{matrix} 5 & - 5 & - 5 \\ - 2 & - 1 & 3 \\ - 1 & 2 & - 1 \end{matrix}]}^{T}$ $= [\begin{matrix} 5 & - 2 & - 1 \\ - 5 & - 1 & 2 \\ - 5 & 3 & - 1 \end{matrix}]$

$\Rightarrow C^{- 1} = \frac{1}{|C|} a d j C$ $= \frac{1}{- 5} [\begin{matrix} 5 & - 2 & - 1 \\ - 5 & - 1 & 2 \\ - 5 & 3 & - 1 \end{matrix}]$

$X = C^{- 1} D$

$\Rightarrow [\begin{matrix} x \\ y \\ z \end{matrix}] = - \frac{1}{5} [\begin{matrix} 5 & - 2 & - 1 \\ - 5 & - 1 & 2 \\ - 5 & 3 & - 1 \end{matrix}] [\begin{matrix} 900 \\ 1600 \\ 2300 \end{matrix}]$

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

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

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

$∴$ $x = 200, y = 300$ and $z = 400$

Hence, the award money for each valueof sincerity, truthfulness and helpfulness is $R s 200$ , $R s 300$ and $R s 400$ .

One more value which should be considered for award hardwork.

Important Questions on Solution of Simultaneous Linear Equations

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