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Module 2/Systems & Gaussian Elimination

Reduced Row-Echelon Form & Gauss–Jordan

Row echelon form gives you a staircase of pivots, but reduced row echelon form (rref) goes further: every pivot is scaled to a leading 1 and stands alone in its column. This lesson covers Definition 5.4, the Gauss–Jordan sweep that produces the rref, and Remark 5.2 — the fact that the rref of a matrix is unique.

Before you start — give these a try

Attempting first primes your brain for the lesson — even if you miss. Nothing is graded or saved; it's just a warm-up.

Is the matrix [120010001]\begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} in reduced row echelon form?

Which statement correctly distinguishes reduced row echelon form (rref) from row echelon form (REF)?

What you’ll be able to do

  • State the three conditions of Definition 5.4 that a matrix must satisfy to be in reduced row echelon form.
  • Carry out Gauss–Jordan elimination: reduce to row echelon form, scale each pivot to a leading 1, then clear every other entry in each pivot column.
  • Use Remark 5.2 to explain why the rref of a matrix is unique even though its row echelon form is not.
  • Determine the rank of a matrix by counting the leading 1's in its rref.
  • Decide whether a given matrix is in reduced row echelon form.

In your course

· MATH2015 · Linear Algebra & Probability
§5.2 Gauss elimination algorithm
  • Definition 5.3Row-echelon form
    A matrix is in row echelon form if all zero rows are at the bottom and the pivot of every nonzero row lies to the right of the pivot of the row above.
  • Definition 5.4Reduced row echelon form
    A matrix is in reduced row echelon form if it is in row echelon form, the leading entry of each nonzero row is a 1, and each column containing a leading 1 has zeros in all its other entries.
  • Remark 5.2Uniqueness of the rref
    The rref may be computed by Gauss–Jordan elimination and is unique; although the REF is not unique, all REFs and the rref of a matrix have the same number of zero rows, with pivots in the same rows and columns.
  • Example 5.4Reduced row echelon form of a 4×4 matrix
    Reduces A to an rref with three leading 1's, giving rank(A) = 3.
Definition 5.4, Remark 5.2, and Example 5.4 appear at the end of §5.2 (the Gauss–Jordan elimination subsection), just before §5.3 on computing the inverse. Definition 5.3 (row echelon form) is the prerequisite introduced earlier in §5.2.
1

From row echelon form to reduced row echelon form

Recall Definition 5.3: a matrix is in row echelon form (REF) when every all-zero row is at the bottom and each pivot (the left-most nonzero entry of a row) lies strictly to the right of the pivot in the row above. Definition 5.4 adds two conditions for reduced row echelon form (rref): (2) the leading entry of every nonzero row equals 1 (a 'leading 1'), and (3) every column containing a leading 1 has zeros in all of its other entries. For instance, [130010]\begin{bmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \end{bmatrix} is in REF with leading 1's but is still not in rref, because column 2 has a nonzero entry (33) above its leading 1. Its rref is [100010]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}.

2

Gauss–Jordan elimination

Gaussian elimination (the forward pass) only makes zeros below each pivot, producing a row echelon form. Gauss–Jordan elimination finishes the job to produce the rref. After reaching REF, work backwards from the last (bottom-right) pivot upward: scale each pivot row so the pivot becomes a leading 1, then add multiples of that row to the rows above to create zeros above the pivot as well. When every pivot is a 1 that stands alone in its column, you have the rref. In short: the forward pass clears entries below the pivots, and the backward pass normalizes the pivots and clears the entries above them.

3

The rref is unique (Remark 5.2)

A matrix has many row echelon forms — different valid choices of row swaps and scalings give different nonzero entries. But Remark 5.2 guarantees it has exactly one reduced row echelon form. Moreover, for a fixed matrix, every REF and the rref share the same number of zero rows, and the pivots always appear in the same rows and columns. The 'skeleton' of the elimination is forced by the matrix itself; driving each pivot column all the way down to a column of the identity removes every remaining ambiguity, leaving one canonical matrix.

4

Reading off the rank

Because pivot positions are invariant (Remark 5.2), the number of pivots is a property of the matrix itself — this number is the rank. Read it off the rref by counting the leading 1's, equivalently the nonzero rows. In Example 5.4 the 4×44 \times 4 matrix reduces to an rref with three leading 1's, so its rank is 3; in Example 5.3 the row echelon form has two pivots, so that matrix has rank 2. For a square n×nn \times n matrix, full rank nn is equivalent to rref=In\mathrm{rref} = I_n.

Definition 5.3 — Row Echelon Form (REF)

A matrix is in row echelon form if: (1) all rows consisting only of zeros are at the bottom; and (2) the leading entry (pivot) — the left-most nonzero entry — of every nonzero row lies to the right of the leading entry of every row above it.

Intuition. The nonzero rows form a descending staircase: each new pivot steps strictly to the right of the one above, and empty (zero) rows are swept to the bottom.
Definition 5.4 — Reduced Row Echelon Form (rref)

A matrix is in reduced row echelon form if: (1) it is in row echelon form; (2) the leading entry in each nonzero row is a 1 (a leading 1); and (3) each column containing a leading 1 has zeros in all its other entries.

Intuition. rref is the most simplified REF: every pivot is normalized to 1 and is the only nonzero number in its column, so each pivot column looks exactly like a column of the identity matrix.
Remark 5.2 — Uniqueness of the rref

The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique. For a given matrix, although the REF is not unique, all row echelon forms and the rref have the same number of zero rows, and the pivots are found in the same rows and columns.

Intuition. You can reach many REFs by different choices, but they all agree on the skeleton: same pivot positions, same number of zero rows. Pushing the simplification all the way with Gauss–Jordan removes every remaining choice, leaving one canonical matrix — the rref.

Worked examples

Example 1

Find the reduced row echelon form of A=[1325]A = \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix}, and state its rank.

  1. 1

    The (1,1)(1,1) entry is already a leading 1, so it is the first pivot. Clear the entry below it: r2→r2−2r1r_2 \to r_2 - 2r_1 gives [130−1]\begin{bmatrix} 1 & 3 \\ 0 & -1 \end{bmatrix}.

  2. 2

    Scale row 2 so its pivot becomes a leading 1: r2→−r2r_2 \to -r_2 gives [1301]\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}. The matrix is now in row echelon form with leading 1's.

  3. 3

    Gauss–Jordan backward sweep: use the row-2 pivot (column 2) to clear the entry above it. r1→r1−3r2r_1 \to r_1 - 3r_2 gives [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.

  4. 4

    Every pivot is a leading 1 and stands alone in its column, so this is the rref.

Answer. rref(A)=[1001]=I2\mathrm{rref}(A) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I_2. There are two leading 1's, so rank(A)=2\mathrm{rank}(A) = 2 (and AA is invertible).
Example 2

(Example 5.4) Find the reduced row echelon form and the rank of A=[2202464756272324]A = \begin{bmatrix} 2 & 2 & 0 & 2 \\ 4 & 6 & 4 & 7 \\ 5 & 6 & 2 & 7 \\ 2 & 3 & 2 & 4 \end{bmatrix}.

Example 3

Is M=[104001]M = \begin{bmatrix} 1 & 0 & 4 \\ 0 & 0 & 1 \end{bmatrix} in reduced row echelon form? If not, finish the reduction.