Evanalysis
2.2Source-backedEmbedded interactionEstimated reading time: 1 min

2.2 Augmented matrices and row operations

Translate a system into `Ax = b`, package it as `[A|b]`, and understand exactly why elementary row operations preserve the solution set.

Note collections

MATH1030: Linear algebra I

Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.

Language

ENEnglish繁Traditional Chinese简Simplified Chinese
All note collectionsSeries overview

Chapter 1

Systems of equations

Learn to read equations as full solution sets.

1.1

1.1 Equations and solution sets

Chapter 2

Matrices and elimination

Build matrix intuition and use row reduction with purpose.

2.1

2.1 Matrix basics

2.2

2.2 Augmented matrices and row operations

2.3

2.3 Gaussian elimination and RREF

2.4

2.4 Solution-set types

Chapter 3

Matrix algebra

Matrix multiplication, transpose, and structural matrix notation.

3.1

3.1 Matrix multiplication and identity matrices

3.2

3.2 Transpose and special matrices

Chapter 4

Solution structure

Homogeneous systems, null spaces, and the shape of full solution sets.

4.1

4.1 Homogeneous systems and null space

Chapter 5

Invertibility

Understand when a matrix can be undone and why that matters.

5.1

5.1 Invertible matrices

Member

Chapter 6

Vector spaces

Move from matrix procedures to the structure of spaces, span, independence, and basis.

6.1

6.1 Vector spaces

Member

6.2

6.2 Subspaces

Member

6.3

6.3 Linear combinations and span

Member

6.4

6.4 Linear dependence and independence

Member

6.5

6.5 Basis and dimension

Member

6.6

6.6 Column space, row space, and rank

Member

A linear system contains two kinds of data: the coefficients attached to the variables, and the constants on the right-hand side. When we solve the system, the variables, plus signs, and equality signs stay in the same pattern; what changes are the numerical entries.

That is why augmented matrices matter. They let us store exactly the data that row operations modify, without rewriting every variable symbol after each step.

Why the augmented matrix is the right package

Consider a system of m linear equations in n unknowns:

a11x1+a12x2+⋯+a1nxn=b1,a21x1+a22x2+⋯+a2nxn=b2, ⋮am1x1+am2x2+⋯+amnxn=bm.\begin{aligned} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= b_1, \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= b_2, \\ &\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= b_m. \end{aligned}

The source notes rewrite this system in three parallel ways:

  • as a list of equations;
  • as a matrix equation Ax=bAx = b;
  • as a single augmented matrix [A|b].

Definition

Coefficient matrix, constant vector, and augmented matrix

For a system Ax=bAx = b:

  • AA is the coefficient matrix;
  • b is the vector of constants;
  • [A|b] is the augmented matrix, formed by placing the constant column beside the coefficient matrix.

The vertical bar is bookkeeping. It reminds you which column came from the right-hand side, but it does not create a new kind of matrix operation.

There is also a useful column-vector reading. If the columns of AA are a1,a2,…,ana_1, a_2, \ldots, a_n, then

Ax=b⟺x1a1+x2a2+⋯+xnan=b.Ax = b \quad\Longleftrightarrow\quad x_1 a_1 + x_2 a_2 + \cdots + x_n a_n = b.

This matters later when we study span and column space. For now, it already explains why a linear system can be read either row by row or column by column.

Worked example

Write one system in all three forms

Consider

x1+2x2+2x3=4,x1+3x2+3x3=5,2x1+6x2+5x3=6.\begin{aligned} x_1 + 2x_2 + 2x_3 &= 4, \\ x_1 + 3x_2 + 3x_3 &= 5, \\ 2x_1 + 6x_2 + 5x_3 &= 6. \end{aligned}

Its coefficient matrix and constant vector are

A=[122133265],b=[456].A = \begin{bmatrix} 1 & 2 & 2 \\ 1 & 3 & 3 \\ 2 & 6 & 5 \end{bmatrix}, \qquad b = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}.

So the system may be written as Ax=bAx = b, where

x=[x1x2x3],x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix},

and its augmented matrix is

[122413352656].\left[ \begin{array}{ccc|c} 1 & 2 & 2 & 4 \\ 1 & 3 & 3 & 5 \\ 2 & 6 & 5 & 6 \end{array} \right].

The three elementary row operations

The local notes define exactly three row operations:

  1. swap two rows;
  2. multiply one row by a nonzero scalar;
  3. add a multiple of one row to another row.

The phrase “nonzero” in operation 2 is essential. Multiplying a row by 0 would erase the equation instead of rewriting it equivalently, so it would no longer be reversible.

Definition

Elementary row operations

The allowed operations on an augmented matrix are:

  1. Ri↔RjR_i \leftrightarrow R_j
  2. Ri←αRiR_i \leftarrow \alpha R_i, where α≠0\alpha \ne 0
  3. Rj←αRi+RjR_j \leftarrow \alpha R_i + R_j

These are called elementary row operations.

Why these operations preserve the solution set

The serious point is not that row operations make a matrix “look nicer.” The serious point is that they preserve the set of solutions.

Theorem

Row-equivalent augmented matrices represent equivalent systems

If one augmented matrix is obtained from another by a sequence of elementary row operations, then the two corresponding systems are equivalent: they have exactly the same solution set.

Proof

Why each elementary row operation is safe

The local notes also stress reversibility. This is worth remembering because it separates genuine equivalence moves from destructive shortcuts.

Read each row operation as an equation operation

When you apply a row operation, do not think “I am changing numbers in a table.” Think “I am rewriting one equation using another equation.”

Worked example

Interpret elimination before you memorize it

Start from

[122413352656].\left[ \begin{array}{ccc|c} 1 & 2 & 2 & 4 \\ 1 & 3 & 3 & 5 \\ 2 & 6 & 5 & 6 \end{array} \right].

Apply

R2←R2−R1,R3←R3−2R1.R_2 \leftarrow R_2 - R_1, \qquad R_3 \leftarrow R_3 - 2R_1.

The new augmented matrix is

[12240111021−2].\left[ \begin{array}{ccc|c} 1 & 2 & 2 & 4 \\ 0 & 1 & 1 & 1 \\ 0 & 2 & 1 & -2 \end{array} \right].

This is not magic. It simply means:

  • equation 2 is replaced by “equation 2 minus equation 1”;
  • equation 3 is replaced by “equation 3 minus 2 times equation 1.”

So the same system is now being read in a form where the first variable is already eliminated from the lower rows. The problem has not changed; only its presentation has changed.

The next note studies Gaussian elimination in detail. At the current stage, the important idea is more basic: an augmented matrix is a compact record of the same system, and row operations are legitimate because they preserve the solution set.

A practical solving strategy

The source notes summarize the matrix method in three steps:

  1. write the system as an augmented matrix;
  2. perform row operations to reach a simpler matrix;
  3. translate the simpler matrix back into equations, or read the answers from a sufficiently simple form.

This strategy is easy to say, but it becomes reliable only when you keep asking what each step is preserving and what structural feature you are trying to create.

Use the explorer below to connect the symbolic row operations with the changing system they represent.

Read and try

Translate one system into a matrix

The live explorer highlights how each equation becomes one matrix row plus one constant entry.

System

  1. x + 2y = 5
  2. 3x - y = 4

Result

125
3-14

Common mistakes and subtle points

Common mistake

The bar in [A|b] is not a wall you may ignore

The last column belongs to the same system. If you apply a row operation to the coefficient entries but leave the constants untouched, you are no longer rewriting the same system.

Common mistake

Multiplying a row by 0 is not an allowed move

The allowed scaling operation requires a nonzero scalar. Multiplying by 0 throws away the equation and is not reversible, so it does not preserve equivalence.

Common mistake

A row operation changes equations, not variables

Row operations combine equations with one another. They do not mean "replace x2x_2 by something else" or "change the meaning of the unknowns."

Quick checks

Quick check

Why is Ri←0RiR_i \leftarrow 0R_i not an allowed row operation?

Answer in terms of reversibility and loss of information.

Solution

Answer

Quick check

If two equations are swapped, does the system get a different solution set?

Do not think about appearance only. Think about what it means to solve all the equations simultaneously.

Solution

Answer

Exercises

Quick check

Write the augmented matrix for the system x1−2x2−x3+x4=1x_1 - 2x_2 - x_3 + x_4 = 1, x2+x3−x4=2x_2 + x_3 - x_4 = 2, x3+2x4=3x_3 + 2x_4 = 3.

Keep the columns in the order x1,x2,x3,x4x_1, x_2, x_3, x_4, even when a variable does not appear in a given equation.

Solution

Guided solution

Quick check

Using the same system, which single row operation eliminates the x2x_2 term from the first equation, and what new first equation do you get?

The coefficient of x2x_2 in equation 1 is −2-2, while the coefficient in equation 2 is 1.

Solution

Guided solution

Read this first

This page builds on 1.1 Equations and solution sets and 2.1 Matrix basics, and it prepares the elimination viewpoint used in 2.3 Gaussian elimination and RREF.

Prerequisites

1.1

1.1 Equations and solution sets

2.1

2.1 Matrix basics

Key terms in this unit

More notes in this series

Previous section

2.1 Matrix basics

Next section

2.3 Gaussian elimination and RREF