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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Reduced Row Echelon Form (RREF)

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Echelon Forms

Echelon forms will be very useful for determining solutions to linear systems.

Row Echelon Form

A matrix is in row echelon form (REF) if:
  • All rows of 0s are at the bottom
  • The first non-zero entry (called a leading entry, leading coefficient, or pivot) in any row is to the right of the leading entries in the rows above it
  • Every entry below a leading entry is 0
Wize Tip
"Échelon" is a French word that means step-ladder.
In row echelon form, the leading entries form a descending step pattern from the top left to the bottom right.

Examples (REF)

[−2−8103−1]\left[\begin{array}{rr|r} -2 & -8 & 1\\ 0 & 3 & -1 \end{array}\right]


[1−1000−2000003]\left[\begin{array}{rrr|r} 1 & -1 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 3 \end{array}\right]


[261−1800−3700005100000]\left[\begin{array}{rrrr|r} 2&6&1&-1&8\\ 0&0&-3&7&0\\ 0&0&0&5&1\\ 0&0&0&0&0 \end{array}\right]

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Reduced Row Echelon Form

A matrix is in reduced row echelon form (RREF) if:
  • It is in row echelon form
  • Every leading entry is a 1
  • In columns with a leading 1, every other entry in the column is 0
Examples (RREF)
[10101−1]\left[\begin{array}{rr|r} 1 & 0 & 1\\ 0 & 1 & -1 \end{array}\right]


[100001000001]\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]


[10008001030001100000]\left[\begin{array}{rrrr|r} 1&0&0&0&8\\ 0&0&1&0&3\\ 0&0&0&1&1\\ 0&0&0&0&0 \end{array}\right]
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Example: Row Echelon Form (REF)

Which of the following matrices are in row echelon form?

1. [440000]\left[\begin{array}{rr|r} 4&4&0\\ 0&0&0 \end{array}\right]

Yes, REF.


2. [440000027]\left[\begin{array}{rr|r} 4&4&0\\ 0&0&0\\ 0&2&7 \end{array}\right]

No, not REF.
The row of 0s should be at the bottom.


3. [4001031800−11]\left[\begin{array}{rrr|r} 4&0&0&1\\ 0&3&1&8\\ 0&0&-1&1 \end{array}\right]

Yes, REF.


4. [−130100210000−32000000]\left[\begin{array}{rrrr|r} -1&3&0&1&0\\ 0&2&1&0&0\\ 0&0&-3&2&0\\ 0&0&0&0&0\\ \end{array}\right]

Yes, REF.


5. [−13010020720002500−300]\left[\begin{array}{rrrr|r} -1&3&0&1&0\\ 0&2&0&7&2\\ 0&0&0&2&5\\ 0&0&-3&0&0\\ \end{array}\right]

No, not REF.
The leading entry of Row 4 [-3] should be right of the leading entries in the rows above, but it is left of the leading entry in Row 3 [2].
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Example: Reduced Row Echelon Form (RREF)

Which of the following matrices are in reduced row echelon form?

1. [1−20000]\left[\begin{array}{rr|r} 1&-2&0\\ 0&0&0 \end{array}\right]

Yes, RREF.


2. [100800150104]\left[\begin{array}{rrr|r} 1&0&0&8\\ 0&0&1&5\\ 0&1&0&4 \end{array}\right]

No, not RREF. (The leading 1 in Row 3 should be right of the leading 1s above, but it is left of the leading 1 in Row 2)


3. [104101180000]\left[\begin{array}{rrr|r} 1&0&4&1\\ 0&1&1&8\\ 0&0&0&0 \end{array}\right]

Yes, RREF.


4. [10010010000012000000]\left[\begin{array}{rrrr|r} 1&0&0&1&0\\ 0&1&0&0&0\\ 0&0&1&2&0\\ 0&0&0&0&0\\ \end{array}\right]

Yes, RREF.


5. [13210010720012500000]\left[\begin{array}{rrrr|r} 1&3&2&1&0\\ 0&1&0&7&2\\ 0&0&1&2&5\\ 0&0&0&0&0\\ \end{array}\right]

No, not RREF. (Columns [vertical lines] with leading 1s should have 0s in all other entries, but Columns 2 and 3 contain non-zero entries)

Practice: REF

Select all matrices that are in row echelon form (REF).

Practice: RREF

Determine the values of aa, bb, and cc so that the following matrix is in reduced row echelon form (RREF).
[a−b30−10a1201+b0c]\left[ \begin{array}{ccc|r} a-b&3&0&-1\\ 0&a&1&2\\ 0&1+b&0&c \end{array} \right]

Practice: RREF

Which of the following is not in row-reduced echelon form (RREF)?


Extra Practice
Find all the values of cc so that the following system of linear equations corresponding to the augmented matrix has a unique solution.
[22410c112c∣221]\left[\begin{array}{l} 2 & 2 & 4 \\ 1 &0 & c \\ 1 & 1 & 2c \\ \end{array}\right.\left|\begin{array}{l} 2\\2\\1\end{array}\right]
Find all solutions to the following system of linear equations[10210131]\left[\begin{array}{rrr|r} 1&0&2&1\\ 0&1&3&1 \end{array}\right].
Example: RREF
Example:
Which of the following are in reduced row echelon form?
a.) [100201000010]\begin{bmatrix} 1 &0 &0& 2\\ 0& 1 & 0 &0\\ 0&0&1&0\\ \end{bmatrix}
The reduced row echelon form of the augmented matrix for a linear system is
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
a) What is the rank of matrix A
How many non-pivot (non-leading) columns can the reduced row echelon form of a non-zero 2×32\times3 matrix have?
Rank of a Matrix
Practice: Rank
In the reduced row echelon form of the coefficient matrix AA that corresponds to a system of linear equations with 6 equations in 9 unknowns, what is the maximum number of leading 1s?
Practice: Row Echelon Matrix
Which of the following correspond with the RREF of the matrix
Find the first row of the row-reduced echelon form of the matrix
A=[123−1−10013]A=\begin{bmatrix}1&2&3\\-1&-1&0\\0&1&3\end{bmatrix}
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
Let the following augmented matrix
[10−5−3017000(k+2)(k−4)(k−4)]\left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have no solution?
[10−5−3017000(k+2)(k−4)(k−4)] \left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have a one-parameter family of solutions (infinitely many solutions)?
[10−5−3017000(k+2)(k−4)(k−4)] \left[\begin{array}{ccc|c} 1 & 0 & -5 & -3\\ 0 & 1 & 7 & 0\\ 0 & 0 & (k+2)(k-4) & (k-4) \end{array} \right]
For what value(s) of kkdoes the system have a unique solution?
Which of the following matrices is/are in row-reduced echelon form?
a.)[101021000]\begin{bmatrix}1&0&1\\ 0&2&1\\0&0&0\end{bmatrix}
b.)[012001]\begin{bmatrix}0&1&2\\ 0&0&1\\\end{bmatrix}
How many non-pivot (non-leading) columns can the reduced row echelon form of a non-zero 2×32\times3 matrix have?
Find all values of aa and bb so that the system of linear equations corresponding to the following augmented matrix has a three-parameter family of solutions.
[1000000a000−1000b00]\left[ \begin{array}{rrrrr|r} 1&0&0&0&0&0\\ 0&a&0&0&0&-1\\ 0&0&0&b&0&0 \end{array} \right]
Writing the solution from RREF
Consider the linear system with unknowns x1,x2,x3x_1,x_2,x_3whose augmented matrix in reduced row echelon form is:
[1−30000100000]\left[ \begin{array}{rrr|r} 1&-3&0&0\\ 0&0&1&0\\ 0&0&0&0 \end{array} \right]
Write the general solution
Given the augmented matrix [1201010200k2−9k−3]\left[\begin{array}{rrr|r} 1 & 2 &0 &1 \\ 0 & 1 &0&2\\ 0&0&k^2-9&k-3 \end{array}\right], answer the following questions.
Find all the values of cc so that the following system of linear equations corresponding to the augmented matrix has a unique solution.
[22410c112c∣221]\left[\begin{array}{l} 2 & 2 & 4 \\ 1 &0 & c \\ 1 & 1 & 2c \\ \end{array}\right.\left|\begin{array}{l} 2\\2\\1\end{array}\right]
Find all solutions to the following system of linear equations[10210131]\left[\begin{array}{rrr|r} 1&0&2&1\\ 0&1&3&1 \end{array}\right].
Find all the values of kk such that the system of linear equations corresponding to the augmented matrix [1201010200k2−9k−3]\left[\begin{array}{rrr|r} 1 & 2 &0 &1 \\ 0 & 1 &0&2\\ 0&0&k^2-9&k-3 \end{array}\right]has infinitely many solutions.
Find all the values of kk such that the system of linear equations corresponding to the augmented matrix [1201010200k2−9k−3]\left[\begin{array}{rrr|r} 1 & 2 &0 &1 \\ 0 & 1 &0&2\\ 0&0&k^2-9&k-3 \end{array}\right] has a unique solution.
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
Reduced Row Echelon Form (RREF)
Which of the following is in RREF (reduced row echelon form)?
A=[1000001001000000]B=[1−100001−100000000]A=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad B=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
C=[1−1000010000−10000]D=[1100011000010000]C=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{bmatrix} \quad\quad D=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Solving Linear Systems
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r} 1&-1&0&0&-5&4\\[0.5em] 0&0&1&0&-2&2\\[0.5em] 0&0&0&1&4&-3 \end{array}\right]
If the unknowns in this system are x1, x2, x3, x4, x5x_1,\ x_2,\ x_3,\ x_4,\ x_5, find the solution(s) to the linear system.
Reduced Row Echelon Form
Which of the following matrices are in row-reduced echelon form? (select all that apply)