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The reduced row echelon form of the augmented matrix for a linear system is [A|…

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

Reduced Row Echelon Form (RREF)

6 Activities

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

Rank of a Matrix

4 Activities

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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

b) Does this linear system has a unique solution, infinity many solutions or no solution?

c) Find all solutions of homogenous system.

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].
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}
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)
More Rank of a Matrix Questions:
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 15
The linear system M‾⋅x⃗=0⃗\bcb{\boldsymbol{ \underline{M} \cdot \vx = \vz}} is known to have only the trivial solution. If M‾\bcb{\boldsymbol{ \underline{M}}} is a matrix with 8 rows and 7 columns, what is its rank?
The 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]_.
Example: Rank of a Matrix
Example:
1. Find the ranks of the following matrices and state the number of solutions.
a.) [102100∣120]\left[\begin{array}{l} 1& 0 \\ 2& 1 \\ 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]
The 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]_.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
A linear system of three equations in four unknowns has
Equivalent Conditions to Full Rank
a) If AA is a 5×35\times 3 matrix of full rank, and BB is a 3×73\times 7 matrix of rank 2, what are the possible values of rank(AB)rank(AB) ?
b) If AA is a 3×53\times 5 matrix of full rank and BB is a 5×45\times 4 matrix of rank 3, what are the possible values of rank(AB)rank(AB) ?
Find the rank of the following matrix:
[525151021200]\begin{bmatrix}5&2&5&1\\5&1&0&2\\1&2&0&0\end{bmatrix}
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b]\left[A\left|b\right.\right]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAis 3, which of the following statements best describes the system?
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b][A|b]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AA is rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AA is 3, which of the following statements best describes the system?
Linear equations
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Matrix of a Projection
a) Find the matrix AA so that for x⃗∈R3\vec x \in \mathbb{R}^3 the transformation mapping x⃗→proj(−13,1,12)x⃗\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x is the same as the transformation mapping x⃗→Ax⃗\vec x\rightarrow A\vec x
b) What is rank(A)rank(A) ?
Invertible Matrix Theorem
Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏} where A5×5A_{5 \times 5} is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])?
Rank of a Matrix
Determine the number of solutions for the system of linear equations Ax⃗=b⃗A\vec{x}=\vec{b} with 5 equations and 5 unknowns, given:
Rank of a Matrix
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, the maximum number of leading 1s is
6
.
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a matrix
Let AA be a 6 x 4 matrix and let Ax=bA\bm{x}=\bm{b} be a system of linear equations. Which of the following statements must always be true? (select all that apply)
Rank of a Matrix
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a Matrix
How many solutions does the system of linear equations corresponding to the following augmented matrix have?
[1010−1101101000011−1000000]\left[ \begin{array}{rrrrr|r} 1&0&1&0&-1&1\\ 0&1&1&0&1&0\\ 0&0&0&1&1&-1\\ 0&0&0&0&0&0 \end{array} \right]
Rank of a Matrix
Which of the following is an augmented matrix corresponding to a system of linear equation that is consistent? (select all that apply)