Two schools A and B want to award their selected teachers on the values of honesty, hard work and regularity. The school A wants to awardRsx each,Rsy each andRsz each for the three respective values to 3, 2 and 1 teachers with a total award money ofRs1.28lakhs. School B wants to spendRs1.54lakhsto award its 4, 1 and 3 teachers on the respective values(by giving the same award money for the three values as before). If the total amount of award for one prize on each value isRs57000, using matrices, find the award money for each value?

 
The 3 equations are
3x+2y+z=1.28
4x+y+3z=.54
x+y+z=0.57


We can calculate matrix as,
A=
  3     2     1  
  4     1     3  
  1     1     1  


Finding the determinant of the matrix А

det A = -5

The determinant of A is non-zero, hence the inverse matrix A-1 exists. To find the inverse matrix was calculated the minors and cofactors of the matrix А



Finding the minor M1,1 and cofactor A1,1. The matrix A cross out the row 1 and column 1.
M1,1
  1     3  
  1     1  
 =  -2
C1,1 = (-1)1+1M1,1 -2

Finding the minor M1,2 and cofactor A1,2. The matrix A cross out the row 1 and column 2.
M1,2
  4     3  
  1     1  
 =  1
C1,2 = (-1)1+2M1,2 -1


Finding the minor M1,3 and cofactor A1,3. The matrix A cross out the row 1 and column 3.
M1,3
  4     1  
  1     1  
 =  3
C1,3 = (-1)1+3M1,3 3


Finding the minor M2,1 and cofactor A2,1. The matrix A cross out the row 2 and column 1.
M2,1
  2     1  
  1     1  
 =  1
C2,1 = (-1)2+1M2,1 -1


Finding the minor M2,2 and cofactor A2,2. The matrix A cross out the row 2 and column 2.
M2,2
  3     1  
  1     1  
 =  2
C2,2 = (-1)2+2M2,2 2


Finding the minor M2,3 and cofactor A2,3. The matrix A cross out the row 2 and column 3.
M2,3
  3     2  
  1     1  
 =  1
C2,3 = (-1)2+3M2,3 -1


Finding the minor M3,1 and cofactor A3,1. The matrix A cross out the row 3 and column 1.
M3,1
  2     1  
  1     3  
 =  5
C3,1 = (-1)3+1M3,1 5


Finding the minor M3,2 and cofactor A3,2. The matrix A cross out the row 3 and column 2.
M3,2
  3     1  
  4     3  
 =  5
C3,2 = (-1)3+2M3,2 -5


Finding the minor M3,3 and cofactor A3,3. The matrix A cross out the row 3 and column 3.
M3,3
  3     2  
  4     1  
 =  -5
C3,3 = (-1)3+3M3,3 -5

Matrix of cofactors:
 
C =
  -2     -1     3  
  -1     2     -1  
  5     -5     -5  


Transpose of cofactor matrix:
 
CT =
  -2     -1     5  
  -1     2     -5  
  3     -1     -5  

Finding the inverse matrix
A-1 =  CT  = 
det A
 
  2/5     1/5     -1  
  1/5     -2/5     1  
  -3/5     1/5     1  

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