29-Jul-2021 01:44:30

rref_test():
  MATLAB/Octave version 9.9.0.1467703 (R2020b)
  rref() computes the reduced row echelon form (RREF) of
  a matrix.

  The RREF is susceptible to roundoff error.
  With that in mind, it can be used to determine:
  * the rank of a square matrix;
  * the column rank of a rectangular matrix;
  * the independent columns of a rectangular matrix
  * if a linear system is consistent;
  * the inverse of a square invertible matrix.

  Test 1, just get the RREF.

  Matrix A:

  Col:       1             2             3             4             5       
  Row
      1 :           1             3             0             2             6  
      2 :          -2            -6             0            -2            -8  
      3 :           3             9             0             0             6  
      4 :          -1            -3             0             1             0  

  Col:       6             7       
  Row
      1 :           3             1  
      2 :           3             1  
      3 :           6             2  
      4 :           9             3  

  A has rank 3

  RREF of A:

  Col:       1             2             3             4             5       
  Row
      1 :           1             3             0             0             2  
      2 :           0             0             0             1             2  
      3 :           0             0             0             0             0  
      4 :           0             0             0             0             0  

  Col:       6             7       
  Row
      1 :           0             0  
      2 :           0             0  
      3 :           1      0.333333  
      4 :           0             0  

  Test 2, look for inconsistency.

  Matrix A:

  Col:       1             2             3             4       
  Row
      1 :           1            -2             3            -1  
      2 :           3            -6             9            -3  
      3 :           0             0             0             0  
      4 :           2            -2             0             1  
      5 :           6            -8             6             0  
      6 :           3             3             6             9  
      7 :           1             1             2             3  

  A has rank 3

  RHS b1:

     1:            1
     2:            2
     3:            3
     4:            4
     5:            5
     6:            5
     7:            7

  [A;b1] has rank 4
  Linear system A*x=b1 is INCONSISTENT.

  RHS b2:

     1:            4
     2:           12
     3:            0
     4:            2
     5:           12
     6:            9
     7:            3

  [A;b2] has rank 3
  Linear system A*x2=b2 is CONSISTENT.

  Solution of A*x2=b2:

     1:            1
     2:            0
     3:            1
     4:            0

  Compare b2 and A*x2:

       1:              4             4
       2:             12            12
       3:              0             0
       4:              2             2
       5:             12            12
       6:              9             9
       7:              3             3

  Test 3, inverse.

  Matrix A:

  Col:       1             2             3             4       
  Row
      1 :           5             7             6             5  
      2 :           7            10             8             7  
      3 :           6             8            10             9  
      4 :           5             7             9            10  

  Estimated A_inv:

  Col:       1             2             3             4       
  Row
      1 :          68           -41           -17            10  
      2 :         -41            25            10            -6  
      3 :         -17            10             5            -3  
      4 :          10            -6            -3             2  

  A_inv * A:

  Col:       1             2             3             4       
  Row
      1 :           1             0             0             0  
      2 :           0             1             0             0  
      3 :           0             0             1             0  
      4 :           0             0             0             1  

  Test 4, independent columns.

  Matrix A:

  Col:       1             2             3             4       
  Row
      1 :           1             1             2             3  
      2 :           2             3             4             9  
      3 :           3             0             6             0  
      4 :           4             2             8             0  
      5 :           5             6            10             6  
      6 :           6             3            12             6  
      7 :           7             1            14             2  

  Number of independent columns is 3

  Independent columns of A:

  Col:       1             2             3       
  Row
      1 :           1             1             3  
      2 :           2             3             9  
      3 :           3             0             0  
      4 :           4             2             0  
      5 :           5             6             6  
      6 :           6             3             6  
      7 :           7             1             2  

rref_test():
  Normal end of execution.

29-Jul-2021 01:44:30