The rref( Command


Command Summary

Puts a matrix into reduced row-echelon form.

Command Syntax


Menu Location


  1. MATRX (on the TI-83) or 2nd MATRX (TI-83+ or higher) to access the matrix menu.
  2. RIGHT to access the math menu.
  3. ALPHA B to select rref(, or use arrows and ENTER.

Calculator Compatibility


Token Size

2 bytes

Given a matrix with at least as many columns as rows, the rref( command puts a matrix into reduced row-echelon form using Gaussian elimination.

This means that as many columns of the result as possible will contain a pivot entry of 1, with all entries in the same column, or to the left of the pivot, being 0.

    [[1 2 5 0]
     [2 2 1 2]
     [3 4 7 3]]
    [[1 0 0 6   ]
     [0 1 0 -5.5]
     [0 0 1 1   ]]

Advanced Uses

The rref( command can be used to solve a system of linear equations. First, take each equation, in the standard form of $\definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen}a_1x_1+\dots + a_nx_n = b$, and put the coefficients into a row of the matrix.

Then, use rref( on the matrix. There are three possibilities now:

  • If the system is solvable, the left part of the result will look like the identity matrix. Then, the final column of the matrix will contain the values of the variables.
  • If the system is inconsistent, and has no solution, then it will end with rows that are all 0 except for the last entry.
  • If the system has infinitely many solutions, it will end with rows that are all 0, including the last entry.

This process can be done by a program fairly easily. However, unless you're certain that the system will always have a unique solution, you should check that the result is in the correct form, before taking the values in the last column as your solution. The Matrâ–ºlist( command can be used to store this column to a list.

Error Conditions

Related Commands


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