
 a) Write the chemical equation :

   a Fe**(2+) + b MnO[4]**(-) + c H**(+) -> d Fe**(3+) + e Mn**(2+) + f H[2]O 
   
 b) Write the list of the chemical elements.
 
         Fe :  a                     =  d                        
         Mn :  b                     =  e            
          O : 4b                     =  f            
          H : c                      =  2f            
        +/- : 2a - b  + c            =  3d + 2e           


 c) Rewrite the list.

         Fe :  a -  d                = 0                        
         Mn :  b -  e                = 0            
          O : 4b -  f                = 0            
          H :  c - 2f                = 0            
        +/- : 2a -  b + c - 3d - 2e  = 0             




 d) Write the list of the chemical elements in a matrix :

           a     b     c     d     e     f     
           -------------------------------
     Fe :  a                -d               = 0                        
     Mn :        b                -e         = 0            
      O :       4b                      -f   = 0            
      H :              c               -2f   = 0            
    +/- : 2a    -b     c   -3d    -2e        = 0             





 e) Complete the matrix :

           a     b     c     d     e     f        
           -------------------------------
     Fe :  1/1  0/1   0/1  -1/1   0/1   0/1  0/1                        
     Mn :  0/1  1/1   0/1   0/1  -1/1   0/1  0/1          
      O :  0/1  4/1   0/1   0/1   0/1  -1/1  0/1          
      H :  0/1  0/1   1/1   0/1   0/1  -2/1  0/1          
    +/- :  2/1 -1/1   1/1  -3/1  -2/1   0/1  0/1           


  f) This system has five rows, and six values to find.
     ( a, b, c, d ,e, f).
     If the system is consistent, it will have many solutions.

     To find a solution, I must give a value, to a coefficient.
     I have chosen to write a = 1/1.

       1,1,  0,1,   0,1,   0,1,   0,1,   0,1,  1,1,    (a = 1/1)
       1,1,  0,1,   0,1,  -1,1,   0,1,   0,1,  0,1,                        
       0,1,  1,1,   0,1,   0,1,  -1,1,   0,1,  0,1,          
       0,1,  4,1,   0,1,   0,1,   0,1,  -1,1,  0,1,          
       0,1,  0,1,   1,1,   0,1,   0,1,  -2,1,  0,1,          
       2,1, -1,1,   1,1,  -3,1,  -2,1,   0,1,  0,1,            

