Answer:
Consider x, y and z are the award of honesty, regularity and hard work and form the system of equations. Then, write them in matrix form as AX = B. Now, the solution is given by \[X={{A}^{-\,1}}B,\] put the values of \[{{A}^{-\,1}},\,\,X\]and B and calculate the required values. Let award for honesty = Rs. x Award for regularity = Rs. y and award for hard work = Rs. z According to first condition, \[x+y+z=6000\] According to second condition, \[3z+x=11000\] According to third condition, \[x+z=2y\] Now, the above equations can be rewritten in standard form of linear equations as \[x+y+z=6000\] ?(i) \[x+0y+3z=11000\] ?(ii) and \[x-2y+z=0\] ?(iii) We can represent these equations using matrices as i.e.\[AX=B\] where, and its solution is given by \[X={{A}^{-\,1}}B\] ?(iv) Now, \[=1\,(0+6)-\,(1-3)+1\,(-\,2-0)\] \[=6+2-2=6\ne 0\] \[\therefore \]\[{{A}^{-\,1}}\]exists, because A is a non-singular matrix. Now, cofactors of \[|A|\]are \[{{A}_{11}}={{(-\,1)}^{1\,\,+\,\,1}}\left| \begin{matrix} 0 & 3 \\ -\,2 & 1 \\ \end{matrix} \right|=+\,\,(0+6)=+\,\,6\] \[{{A}_{12}}={{(-\,1)}^{1\,\,+\,\,2}}\left| \begin{matrix} 1 & 3 \\ 1 & 1 \\ \end{matrix} \right|=-\,(1-3)=2\] \[{{A}_{13}}={{(-\,1)}^{1\,\,+\,\,3}}\left| \begin{matrix} 1 & 0 \\ 1 & -\,2 \\ \end{matrix} \right|=+\,(-\,2)=-\,2\] \[{{A}_{21}}={{(-\,1)}^{2\,\,+\,\,1}}\left| \begin{matrix} 1 & 1 \\ -\,2 & 1 \\ \end{matrix} \right|=-\,(1+2)=-\,3\] \[{{A}_{22}}={{(-\,1)}^{2\,\,+\,\,2}}\left| \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right|=+\,(1-1)=0\] \[{{A}_{23}}={{(-\,1)}^{2\,\,+\,\,3}}\left| \begin{matrix} 1 & 1 \\ 1 & -\,2 \\ \end{matrix} \right|=-\,(-\,2-1)=3\] \[{{A}_{31}}={{(-\,1)}^{3\,\,+\,\,1}}\left| \begin{matrix} 1 & 1 \\ 0 & 3 \\ \end{matrix} \right|=+\,(3-0)=3\] \[{{A}_{32}}={{(-\,1)}^{3\,\,+\,\,2}}\left| \begin{matrix} 1 & 1 \\ 1 & 0 \\ \end{matrix} \right|=-\,(3-1)=-\,2\] \[{{A}_{33}}={{(-\,1)}^{3\,\,+\,\,3}}\left| \begin{matrix} 1 & 1 \\ 1 & 0 \\ \end{matrix} \right|=+\,(0-1)=-\,1\] Now \[\therefore \] Then, Now, \[\Rightarrow \] On comparing, we get \[x=500\], \[y=2000\]and \[z=3500\] Hence, award for honesty = Rs. 500 Award for regularity = Rs. 2000 and award for hard work = Rs. 3500 Value The school must include punctuality for award.
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