Question

given the initial simplex tableau (Matrix):

x              y              s1            s2            p
6              9              1              0              0              300
5              4              0              1              0              180
-3            -4            0              0              1              0

Show the matrices produced by each pivot

Answer 1

Given initial tableau or matrix and we need to find the further matrices by finding pivot elements

x     y      s1    s2     p            

6      9      1      0      0      300   

5      4      0      1      0      180   

-3     -4     0      0      1      0      

Here the most negative element in the bottom row will indicates the pivot element so here -4 ,so we have in column 2 so I am taking 2^nd column as a pivot column and for pivot row the least positive result when last column divided by pivot column will indicates so

i.e. +min (300/9 , 180/4) = 300/9 so 1^st row as a pivot row.

R₁-> R₁ (1/9)

   x      y      s1      s2     p            

2/3    1      1/9 0      0      100/3

   5      4      0        1      0      180   

-3     -4     0        0      1      0

R₂-> R₂ - 4R₁        R₃-> R₃ + 4R₁     

x      y    s1    s2     p            

2/3    1      1/9    0      0      100/3

7/3    0      -4/9   1      0      140/3

-1/3   0      4/9    0      1      400/3

Here the most negative element in the bottom row will indicates the pivot element so here –1/3 ,so we have in column 1   so I am taking 1^ST    column as a pivot column for pivot row the least positive result when last column divided by pivot column will indicates so

i.e. +min ((100/3)/(2/3) , ((140/3)/(7/3)) = (140/3)/(7/3) so 2^nd    row as a pivot row.

R₂-> R₂ (3/7)

x       y        s1      s2     p            

2/3    1 1/9      0      0      100/3

1        0     -4/21 3/7    0      20   

-1/3   0 4/9      0      1      400/3

R₁-> R₁ - (2/3) R₂                R₃-> R₃ + (1/3)R₂

x      y    s1     s2      p            

0      1      5/21   -2/7    0      20    

1      0     -4/21    3/7    0      20    

0      0      8/21     1/7    1      140   

So the above tableau or matrix is the final matrix