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13-May-2025 17:17:22

rref_test():
  MATLAB/Octave version 9.11.0.2358333 (R2021b) Update 7
  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 solution of a nonsingular square linear system;
  * the inverse of a nonsingular square matrix.
  * the determinant of a square matrix.
  * the column rank of a rectangular matrix;
  * the independent columns of a rectangular matrix
  * if a linear system is consistent;

  Test 1, just get the RREF.

  Matrix A:
     1     3     0     2     6     3     1
    -2    -6     0    -2    -8     3     1
     3     9     0     0     6     6     2
    -1    -3     0     1     0     9     3


  A has rank 3

  RREF of A:
    1.0000    3.0000         0         0    2.0000         0         0
         0         0         0    1.0000    2.0000         0         0
         0         0         0         0         0    1.0000    0.3333
         0         0         0         0         0         0         0


  Test 2, look for inconsistency.

  Matrix A:
     1    -2     3    -1
     3    -6     9    -3
     0     0     0     0
     2    -2     0     1
     6    -8     6     0
     3     3     6     9
     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:
     5     7     6     5
     7    10     8     7
     6     8    10     9
     5     7     9    10


  Estimated inverse of A:
    68   -41   -17    10
   -41    25    10    -6
   -17    10     5    -3
    10    -6    -3     2


  Product A_inv * A:
     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1


  Test 4, determinant.

  Matrix A:
     5     7     6     5
     7    10     8     7
     6     8    10     9
     5     7     9    10


  Estimated determinant of A = 1:

  MATLAB det(A) = 1

  Test 5, independent columns.

  Matrix A:
     1     1     2     3
     2     3     4     9
     3     0     6     0
     4     2     8     0
     5     6    10     6
     6     3    12     6
     7     1    14     2


  Number of independent columns is 3

  Independent columns of A:
     1     1     3
     2     3     9
     3     0     0
     4     2     0
     5     6     6
     6     3     6
     7     1     2


rref_test():
  Normal end of execution.

13-May-2025 17:17:23