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10.9.17  Row reduction to echelon form in ℤ/p

The rref command finds the reduced row echelon form of a matrix with elements in ℤ/pℤ (see 14.7.3).

Example

rref([[0,2,9]%15,[1,10,1]%15,[2,3,4]%15])
     



    1%150%150%15
    0%151%150%15
    0%150%151%15



          

This can be used to solve a linear system of equations with coefficients in ℤ/pℤ by rewriting it in matrix form

A X=B.     (1)

rref can then take as argument the augmented matrix of the system (the matrix obtained by augmenting matrix A to the right with the column vector B).

rref returns a matrix [A1,B1] where A1 has ones on its principal diagonal and zeros outside. The solutions in ℤ/pℤ of:

  A1 X=B1

are the same as the solutions of (1).

Example

Solve in ℤ/13ℤ:

  

      x+2y=9,
      3x+10y=0.
rref([[1,2,9]%13,[3,10,0]%13])

or:

rref([[1,2,9],[3,10,0]])%13
     


    1%130%133%13
    0%131%133%13


          

hence x=3%13 and y=3%13.


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