High School
SAT
ACT
University
MCAT
LSAT
DAT

Example: RREF

Related Topics

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

Reduced Row Echelon Form (RREF)

6 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

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}


b.) [123000000]\begin{bmatrix} 1 &2 &3 \\ 0& 0 & 0 \\ 0& 0 & 0 \\ \end{bmatrix}


c.) [12000120]\begin{bmatrix} 1 & 2 & 0&0 \\ 0 & 1 & 2&0 \\ \end{bmatrix}


d.) [12000012]\begin{bmatrix}1&2&0&0\\0&0&1&2\end{bmatrix}
e.) [100001010]\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\\\end{bmatrix}


f.) [0020]\begin{bmatrix}0&0\\2&0\end{bmatrix}


g.) [0000]\begin{bmatrix}0&0\\0&0\end{bmatrix}


h.) [000000001]\begin{bmatrix}0&0&0\\0&0&0\\0&0&1\end{bmatrix}

i.) [1]\begin{bmatrix}1\end{bmatrix}


j.) [0]\begin{bmatrix}0\end{bmatrix}
More Reduced Row Echelon Form (RREF) 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].
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)