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### Answers (1)

Answer:

\begin{aligned} x=200, y=300, z=400 \end{aligned}

Given:

According to question

\begin{aligned} &x+y+z=9000 \\ &3 x+2 y+z=1600 \\ &4 x+y+3 z=2300 \end{aligned}

Hint:

X=A-1B is used to solve this problem.

Ans the determinant and co-factor of matrix A, take it’s transpose that will be Adj A using Adj A calculate A-1.

Solution:

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

Since, the total cash award is Rs900

\begin{aligned} &x+y+z=900\; \; \; \; \; \; ......(1) \\ \end{aligned}

Award money given by school A is Rs1600

\begin{aligned} &3 x+2 y+z=1600\; \; \; \; \; \; ........(2) \end{aligned}

Award money given by school B is Rs2300

\begin{aligned} 4 x+y+3 z=2300\; \; \; \; \; \; .........(3) \end{aligned}

From(1),(2)&(3)

\begin{aligned} &{\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]} \\ &A X=B \end{aligned}

\begin{aligned} |A|=\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 \neq 0 \end{aligned}

\begin{aligned} &c_{11}=(-1)^{1+1}\left|\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right|=5 \\ &c_{12}=(-1)^{1+2}\left|\begin{array}{ll} 3 & 1 \\ 4 & 3 \end{array}\right|=-5 \\ &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 \end{aligned}

\begin{aligned} &c_{22}=(-1)^{2+2}\left|\begin{array}{ll} 1 & 1 \\ 4 & 3 \end{array}\right|=-1 \\ &c_{23}=(-1)^{2+13}\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 \\ &c_{32}=(-1)^{3+2}\left|\begin{array}{ll} 1 & 1 \\ 3 & 1 \end{array}\right|=2 \\ &c_{33}=(-1)^{3+3}\left|\begin{array}{ll} 1 & 1 \\ 3 & 2 \end{array}\right|=-1 \end{aligned}

\begin{aligned} &\operatorname{adj} C=\left[\begin{array}{ccc} 5 & -5 & -5 \\ -2 & -1 & 3 \\ -1 & 2 & -1 \end{array}\right]^{T}=\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ &C^{-1}=\frac{1}{|A|} a d j C \end{aligned}

\begin{aligned} &=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ x=& C^{-1} B \end{aligned}

\begin{aligned} \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=& \frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right]\left[\begin{array}{c} 900 \\ 1600 \\ 2300 \end{array}\right] \\ &=-\frac{1}{5}\left[\begin{array}{c} 4500-3200-2300 \\ -4500-1600+4600 \\ -4500+4800-2300 \end{array}\right] \end{aligned}

\begin{aligned} =-\frac{1}{5}\left[\begin{array}{l} -1000 \\ -1500 \\ -2000 \end{array}\right] \\ \left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 200 \\ 300 \\ 400 \end{array}\right] \\ x=200, y=300, z=400 \end{aligned}

Hence the award money for each value of sincerity, truthfulness and helpfulness is 200, 300 & 400 one more value which should be considered for award hard work

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