18.2. ROW OPERATIONS AND LINEAR EQUATIONS 389

If you want it in terms of fractions, you do the following.>>A=[1 2 3 4;2 3 -11 12;3 5 6 7];rref(sym(A))ans =[ 1, 0, 0, -111/14][ 0, 1, 0, 97/14][ 0, 0, 1, -9/14]You need to have the symbolic toolbox installed with MATLAB for this option.

18.2.2 Uniqueness

I keep referring to the row reduced echelon form. Is there only one? This would besurprising given the infinitely many ways of doing row operations. However, it is in factthe case. Any two sequences of row operations which yield a matrix in row reduced echelonform give the same thing.

Corollary 18.2.22 The row reduced echelon form is unique. That is if B,C are twomatrices in row reduced echelon form and both are row equivalent to A, then B =C.

Proof: Suppose B and C are both row reduced echelon forms for the matrix A. Thenthey clearly have the same zero columns since row operations leave zero columns un-changed. In reading from left to right in B, suppose e1, · · · ,er occur first in positionsi1, · · · , ir respectively. The description of the row reduced echelon form means that each ofthese columns is not a linear combination of the preceding columns. Therefore, by Lemma18.2.18, the same is true of the columns in positions i1, i2, · · · , ir for C. It follows from thedescription of the row reduced echelon form that in C, e1, · · · ,er occur first in positionsi1, i2, · · · , ir. Therefore, both B and C have the sequence e1,e2, · · · ,er occurring first in thepositions, i1, i2, · · · , ir. By Lemma 18.2.18, the columns between the ik and ik+1 positionin the two matrices are linear combinations involving the same scalars of the columns inthe i1, · · · , ik position. Also the columns after the ir position are linear combinations ofthe columns in the i1, · · · , ir positions involving the same scalars in both matrices. Thisis equivalent to the assertion that each of these columns is identical and this proves thecorollary.

Definition 18.2.23 If A is an n× n matrix, and e1, · · · ,er occur for the first timewhen viewed from left to right in positions i1, · · · , ir, then columns i1, · · · , ir in the originalmatrix A are called pivot columns. The rank of the matrix A is the number of these pivotcolumns.

From the description of the row reduced echelon form, every column in this matrix is alinear combination of the pivot columns. Therefore, from Lemma 18.2.18 the same is truefor the columns of the original matrix A.

Example 18.2.24 Let A =

 1 2 3 01 2 3 2−3 2 −1 4

 . Identify the pivot columns and rank.