Example 4

Consider a system of 5 equations in 4 unknowns, gx=h, where g is 5x4.

In[17]:=

  Clear[g,h]
  g={{2,1,1,-2},{3,-2,1,-6},{1,1,-1,-1},{6,0,1,-9},{5,-1,2,-8}};
  h={{1},{-2},{-1},{-2},{3}};
  MatrixForm[g]
  MatrixForm[h]

Out[17]=

  2    1    1    -2
  
  3    -2   1    -6
  
  1    1    -1   -1
  
  6    0    1    -9
  
  5    -1   2    -8

Out[18]=

  1
  
  -2
  
  -1
  
  -2
  
  3

In[19]:=

  gh=AppendRows[g,h];
  MatrixForm[gh]

Out[19]=

  2    1    1    -2   1
  
  3    -2   1    -6   -2
  
  1    1    -1   -1   -1
  
  6    0    1    -9   -2
  
  5    -1   2    -8   3

In[20]:=

  MatrixForm[RowReduce[gh]]

Out[20]=

                            17
                          -(--)
  1       0       0         11    0
  
                          9
                          --
  0       1       0       11      0
  
                          3
                          --
  0       0       1       11      0
  
  
  
  0       0       0       0       1
  
  
  
  0       0       0       0       0

The last row of zeros presents no problems. The second last row is troublesome. It says
0x1+0x2+0x3+0x4=1, which is impossible. This system has no solutions.

Notice that the last column is a pivot column!!!

I do not want to leave you with the impression that the size of the matrix determines whether or not we have a solution or the number of solutions. Consider the following:

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