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Command Summary

Puts a matrix into 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 submenu.
  3. ALPHA A to select ref(, or use arrows.

Calculator Compatibility


Token Size

2 bytes

Given a matrix with at least as many columns as it has rows, the ref( command uses a technique called Gaussian elimination to put the matrix into row-echelon form.

This means that the leftmost N columns (if the matrix has N rows) of the matrix are upper triangular - all entries below the main diagonal are zero. What's more, every entry on the main diagonal is either 0 or 1.

	[[1 2 5 0]
	 [2 2 1 2]
	 [3 4 6 2]
	[[1 4/3 2   2/3]
	 [0 1   9/2 -1 ]
	 [0 0   0   0  ]]

Advanced Uses

In theory, a system of linear equations in N variables can be solved using the ref( command - an equation of the form <math>a_1x_1+\dots + a_nx_n = b</math> becomes a row <math>a_1, \dots, a_n, b</math>, and is put into the matrix. If there is a sufficient number of conditions, the last row of the reduced matrix will give you the value of the last variable, and back-substitution will give you the others.

In practice, it's easier to use Rref( instead for the same purpose.

Error Conditions

Related Commands

  • Rref(
  • |- and other row operations.