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RREF &
Solution Interpretation

Learn how to reduce augmented matrices to Reduced Row Echelon Form and interpret the solutions: unique, infinite, or none.

Linear Algebra

What Is an Augmented Matrix?

A system of linear equations can be compactly written as a matrix. The augmented matrix places the coefficient matrix side by side with the constants column, separated by a vertical bar.

Row operations on this matrix are equivalent to algebraic manipulations on the original equations: the solutions stay the same.

From REF to RREF

Row Echelon Form (REF)
All zero rows at bottom. Each leading entry is to the right of the one above. Entries below each leading entry are zero.
Reduced REF (RREF)
All REF conditions, plus: each leading entry is 1, and it is the only nonzero entry in its column.

The Three Row Operations

1
Swap two rows: \( R_i \leftrightarrow R_j \)
2
Scale a row by a nonzero constant: \( R_i \to c \cdot R_i \), where \( c \neq 0 \)
3
Replace a row with itself plus a multiple of another: \( R_i \to R_i + c \cdot R_j \)

The RREF Criteria

1
All zero rows are at the bottom of the matrix.
2
The leading entry (first nonzero entry) of each nonzero row is 1: called a pivot.
3
Each pivot is in a column to the right of the pivot in the row above.
4
Each pivot is the only nonzero entry in its column (zeros above and below).

Visualizing Pivot Positions

\(1\)
\(0\)
\(0\)
 ★
\(3\)
\(0\)
\(1\)
\(0\)
 ★
\(-1\)
\(0\)
\(0\)
\(1\)
 ★
\(5\)
 pivot pivot pivot free

Pivot vs Free Variables

Pivot Variable Free Variable
Column contains A leading 1 No leading 1
Value determined? Yes: expressed in terms of free vars No: can be any real number
In unique solution Every variable is pivot None exist
In infinite solutions At least one exists At least one exists

Three Types of Solutions

Unique Solution
Every column (except the last) is a pivot column. No free variables. The system has exactly one solution.
Infinitely Many
The system is consistent but has at least one free variable. Solutions form a line, plane, or higher-dimensional set.
No Solution
A row of the form \( \left[\begin{array}{ccc|c} 0 & 0 & \cdots & b \end{array}\right] \) with \( b \neq 0 \) appears. The system is inconsistent.

How to Apply RREF

1
Write the augmented matrix from the system of equations.
2
Reduce using row operations: create pivots left to right, top to bottom. Make each pivot 1, then clear above and below.
3
Check for inconsistency: any row \( [0\ 0\ \cdots\ 0\ |\ b] \) with \( b \neq 0 \)? If yes: no solution.
4
Identify pivot columns vs free columns. Pivot columns → basic variables; non-pivot columns → free variables.
5
Write the solution: if no free variables → unique. If free variables exist → express basic variables in terms of free ones.

Example 1: Unique Solution

Solve the system: \( x + y + z = 6 \), \( 2x + 3y + z = 14 \), \( x - y + 2z = 2 \).

Example 2: Infinitely Many Solutions

Solve: \( x + 2y - z = 3 \), \( 2x + 4y - 2z = 6 \).

Example 3: No Solution

Solve: \( x + y = 2 \), \( x + y = 5 \).

Geometrically, two parallel lines with different intercepts never meet.

Writing Parametric Solutions

When free variables exist, express the solution as a vector plus parameter times direction:

Rank and the Big Picture

The rank of a matrix is the number of pivots in its RREF. For an \( m \times n \) coefficient matrix:

\( \text{rank} = n \)
Every variable is basic. If consistent: unique solution.
\( \text{rank} < n \)
\( n - \text{rank} \) free variables. If consistent: infinitely many solutions.

Common Mistakes

✕ Stopping at REF
REF has zeros below pivots but not above. RREF requires zeros above and below each pivot. Finish the job.
✕ Ignoring Inconsistency
A row \( [0\ 0\ \cdots\ |\ b] \) with \( b \neq 0 \) means no solution. Don't try to assign free variables if the system is inconsistent.
✕ Forgetting Free Variables
If a column has no pivot, its variable is free. Skipping it leads to an incomplete or wrong answer.

Summary Table

Unique Infinite None
RREF pattern Every variable column is a pivot column Non-pivot columns exist, consistent \( \left[0\ \cdots\ 0\ |\ b\right], b \neq 0 \)
Free variables 0 1 or more N/A
rank vs \( n \) \( \text{rank} = n \) \( \text{rank} < n \) \( \text{rank(coeff)} < \text{rank(aug)} \)
Geometric picture Single point Line, plane, etc. Parallel (no intersection)

Question 1 of 5

Which of the following is in Reduced Row Echelon Form?

Question 2 of 5

The RREF of an augmented matrix is
\( \left[\begin{array}{ccc|c} 1 & 0 & 3 & 7 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] \).
How many free variables does this system have?

Question 3 of 5

If a \( 3 \times 4 \) augmented matrix (3 equations, 3 unknowns) reduces to the RREF
\( \left[\begin{array}{ccc|c} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 5 \end{array}\right] \),
what is the solution?

Question 4 of 5

A system's RREF is \( \left[\begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 1 & -2 \end{array}\right] \). What type of solution does it have?

Question 5 of 5

In the RREF \( \left[\begin{array}{cccc|c} 1 & 3 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & 4 \end{array}\right] \), which variable(s) are free?