Skip to content
VidiMaster it, module by module
Module 2/Systems & Gaussian Elimination

Elementary Row Operations & Elementary Matrices

Master the three elementary row operations, encode each one as an elementary matrix, and see Theorem 5.2 turn every row operation into a single left-multiplication A→EAA \to EA — the engine behind Gaussian elimination.

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.

Apply 'add −2-2 times row 1 to row 2' (−2r1+r2-2r_1+r_2) to A=[132251014]A=\begin{bmatrix}1&3&2\\2&5&1\\0&1&4\end{bmatrix}. Enter the result.

Write the 3×33 \times 3 elementary matrix EE that multiplies row 2 by 5 (so that EAEA scales the second row of any 3×33 \times 3 matrix AA by 5).

What you’ll be able to do

  • State the three elementary row operations of Definition 5.1: interchange two rows; scale a row by λ≠0\lambda \neq 0; add λ\lambda times one row to another.
  • Build the elementary matrix (Definition 5.2) for any given row operation by applying that operation to the identity ImI_m.
  • Use Theorem 5.2 to perform a row operation as a left-multiplication A→EAA \to EA by the matching elementary matrix.
  • Explain, via Theorem 5.1, why elementary row operations never change the row rank of a matrix.
  • Recall Remark 5.1 — every elementary matrix has full rank and is therefore invertible — and write the inverse (reverse) operation.

In your course

· MATH2015 · Linear Algebra & Probability
§5.1 Elementary Row Operations & Elementary Matrices
  • Definition 5.1Elementary row transformations
    The following operations are elementary row transformations: (1) interchange two rows; (2) multiply a row by a scalar λ≠0\lambda \neq 0; (3) add λ\lambda times row rjr_j to row rir_i (i≠ji \neq j).
  • Theorem 5.1Row operations preserve row rank
    Elementary row transformations do not change the row rank of a matrix.
  • Definition 5.2Elementary matrix
    The elementary matrix associated with an elementary row operation for m×nm \times n matrices is the m×mm \times m matrix obtained by applying the row operation to the identity matrix ImI_m.
  • Theorem 5.2Row operation = left-multiplication
    An elementary row transformation of an m×nm \times n matrix AA corresponds to multiplication on the left by the corresponding m×mm \times m elementary matrix.
  • Remark 5.1Elementary matrices are invertible
    Elementary matrices have full rank and are therefore invertible.
From §5.1, illustrated by Examples 5.1–5.2. Definition 5.1 phrases operation 3 as 'add λrj\lambda r_j to rir_i', while Theorem 5.2 lists it as 'add λ\lambda times row ii to row jj'; both describe the same move, and the scalar λ\lambda always lands in the (target row, source row) entry of the elementary matrix. This section sets up the Gauss elimination algorithm in §5.2.
1

The three elementary row operations (Definition 5.1)

Gaussian elimination is built from just three moves on the rows of a matrix, called elementary row transformations. For a matrix with rows r1,…,rmr_1,\dots,r_m:

  1. Interchange two rows: ri↔rjr_i \leftrightarrow r_j.
  2. Multiply a row by a scalar λ≠0\lambda \neq 0: ri→λrir_i \to \lambda r_i.
  3. Add λ\lambda times one row to another (with i≠ji \neq j): ri→ri+λrjr_i \to r_i + \lambda r_j.

The nonzero-scalar condition in move 2 is essential: multiplying a row by 00 would destroy information (and can lower the rank), so it is not allowed.

2

Elementary matrices (Definition 5.2)

An elementary matrix is the m×mm \times m matrix you get by applying a single elementary row operation to the identity matrix ImI_m. There is one type for each operation. Written out for the 3×33 \times 3 case:

Swap rows 1 and 2: [010100001]\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix} Multiply row 2 by λ\lambda: [1000λ0001]\begin{bmatrix}1&0&0\\0&\lambda&0\\0&0&1\end{bmatrix} Add λ\lambda times row 1 to row 2: [100λ10001]\begin{bmatrix}1&0&0\\\lambda&1&0\\0&0&1\end{bmatrix}

In the third type the scalar λ\lambda sits in the (target row, source row) entry.

3

A row operation is a left-multiplication (Theorem 5.2)

Every elementary row operation on an m×nm \times n matrix AA can be carried out by multiplying AA on the left by the corresponding m×mm \times m elementary matrix EE. Symbolically, A→EAA \to EA, where EE is that same operation applied to ImI_m.

This does not speed up computation by hand, but it is a powerful proof tool, and it is the key idea behind finding a matrix inverse by row reduction later in the chapter.

4

Rank is preserved; elementary matrices are invertible (Theorem 5.1 & Remark 5.1)

Theorem 5.1 says elementary row operations do not change the row rank of a matrix. The reason: none of the three moves changes the row space span⁡(r1,…,rm)\operatorname{span}(r_1,\dots,r_m) — reordering rows, scaling a row by λ≠0\lambda \neq 0, or adding a multiple of one row to another all leave the span unchanged, so the number of linearly independent rows stays the same. Hence row-reducing to echelon form is a safe way to read off the rank.

Remark 5.1 adds that every elementary matrix has full rank and is therefore invertible; its inverse is again an elementary matrix — the one that undoes the operation (for instance, the inverse of 'add λrj\lambda r_j to rir_i' is 'add −λrj-\lambda r_j to rir_i').

Theorem 5.1 — Invariance of row rank

Elementary row transformations do not change the row rank of a matrix.

Intuition. Each of the three operations leaves the row space span⁡(r1,…,rm)\operatorname{span}(r_1,\dots,r_m) untouched — swapping only reorders the generators, scaling by λ≠0\lambda \neq 0 keeps the same line, and adding a multiple of one row to another is reversible — so the maximum number of linearly independent rows (the rank) cannot change.
Theorem 5.2 — Row operation as left-multiplication

An elementary row transformation of an m×nm \times n matrix AA corresponds to multiplication on the left by the corresponding m×mm \times m elementary matrix EE; the result is EAEA.

Intuition. Left-multiplying by a matrix forms new rows as combinations of AA's rows. If EE is ImI_m with exactly one row operation applied, those combinations reproduce precisely that operation on AA.
Remark 5.1 — Elementary matrices are invertible

Every elementary matrix has full rank and is therefore invertible.

Intuition. An elementary matrix comes from ImI_m (full rank) via a rank-preserving operation (Theorem 5.1), so it still has full rank. Its inverse is the elementary matrix of the reverse operation: undo a swap by swapping back, undo λri\lambda r_i by 1λri\tfrac{1}{\lambda} r_i, and undo adding λrj\lambda r_j by subtracting it.

Worked examples

Example 1

Apply each of the three elementary row operations to A=[123456789]A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix} (Example 5.1).

  1. 1

    Operation 1 — interchange rows 1 and 2 (r1↔r2r_1 \leftrightarrow r_2): swap the first two rows to get [456123789]\begin{bmatrix}4&5&6\\1&2&3\\7&8&9\end{bmatrix}.

  2. 2

    Operation 2 — multiply row 2 by 3 (3r23r_2): 3⋅[4  5  6]=[12  15  18]3\cdot[4\;5\;6]=[12\;15\;18], giving [123121518789]\begin{bmatrix}1&2&3\\12&15&18\\7&8&9\end{bmatrix}.

  3. 3

    Operation 3 — add −4-4 times row 1 to row 2 (−4r1+r2-4r_1+r_2): −4⋅[1  2  3]+[4  5  6]=[0  −3  −6]-4\cdot[1\;2\;3]+[4\;5\;6]=[0\;{-3}\;{-6}], giving [1230−3−6789]\begin{bmatrix}1&2&3\\0&-3&-6\\7&8&9\end{bmatrix}.

Answer. The three results are [456123789]\begin{bmatrix}4&5&6\\1&2&3\\7&8&9\end{bmatrix}, [123121518789]\begin{bmatrix}1&2&3\\12&15&18\\7&8&9\end{bmatrix}, and [1230−3−6789]\begin{bmatrix}1&2&3\\0&-3&-6\\7&8&9\end{bmatrix}.
Example 2

Redo each operation of Example 5.1 as a left-multiplication by an elementary matrix (Example 5.2), with A=[123456789]A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}.

Example 3

The elementary matrix E=[100−410001]E=\begin{bmatrix}1&0&0\\-4&1&0\\0&0&1\end{bmatrix} performs 'add −4-4 times row 1 to row 2'. Find E−1E^{-1} and the operation it performs (illustrating Remark 5.1).