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Find all the values of k such that the system of linear equations corresponding…

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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)

Augmented Matrix

5 Activities

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.


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}
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.
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 Augmented Matrix Questions:
Identify whether the following are systems of linear equations. For those that are linear, write
the corresponding augmented matrix.
a.    2𝑥1−𝑥3=𝑥2a.\ \ \ \ 2𝑥_1−𝑥_3=𝑥_2
Writing the augmented matrix of a system of linear equations
Write the augmented matrix for the following system:
−x1+2x3=5−x25x1+x2=9+x310x3+2x2=1−x1\begin{array}{rcr} -x_1+2x_3&=&5-x_2\\ 5x_1+x_2&=&9+x_3\\ 10x_3+2x_2&=&1-x_1 \end{array}
Writing the augmented matrix of a system of linear equations
Write the augmented matrix for the following system:
x1+6x2−x3−5x4=3x1−x2+4x4=6−3x2−3x3+x4=02x1+3x2−2x3+x4=−1\begin{array}{rcr} x_1+6x_2-x_3-5x_4&=&3\\ x_1-x_2+4x_4&=& 6\\ -3x_2-3x_3+x_4&=&0\\ 2x_1+3x_2-2x_3+x_4&=&-1 \end{array}
Practice: Augmented Matrix
Write the augmented matrix that represents this system of linear equations
x1+x2=7−3x32−x1=7x1+x2+x3=0\begin{array}{rcl} x_1+x_2&=&7-3x_3\\ 2-x_1&=&7\\ x_1+x_2+x_3&=&0 \end{array}
Solving a system of linear equations
Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.
Augmented Matrix
Convert the following SLE into an augmented matrix.
𝑥−𝑦=5𝑧+7x2−z=30=4−x−8y\begin{array}{} 𝑥−𝑦=5𝑧+7\\ \frac{x}{2}-z=3\\ 0=4-x-8y \end{array}
123 - FML #1 - 18.1W - e.g. 20
Consider the following system of linear equations:
{x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6\bcb{ \begin{cases} x - 2y + 4z = 2& \\ 2x + y - 2z = -1 &\\ 3x - y + 2z = 1 &\\ 2x + 6y - 12z = -6& \end{cases}}
a) Construct the augmented matrix A\bcb{A} corresponding to this system.
Which of the following represents the systems of linear equations corresponding to the following augmented matrices?
[11−302520−4∣630]\left[\begin{array}{l} 1 & 1 & -3 \\ 0 &2 & 5 \\ 2 & 0 & -4 \\ \end{array}\right.\left|\begin{array}{l} 6\\3\\0\end{array}\right]
Additional Practice Problems--Systems of Linear Equations
1. Write the augmented matrix that represents this system of linear equations
x1+x2=7−3x32−x1=7x1+x2+x3=0\begin{array}{rcl} x_1+x_2&=&7-3x_3\\ 2-x_1&=&7\\ x_1+x_2+x_3&=&0 \end{array}
2. Determine all values of a, ba,\ b and cc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r} 2&-1&0\\ 0&1&-3 \end{array}\right] represents the following system of linear equations: ax1=bx2(a+c)x2+3=0\begin{array}{rcl} ax_1&=&bx_2\\ (a+c)x_2+3&=&0 \end{array}
Consider a homogeneous system AX=0AX=0 where AA is a 2×42\times4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=0 is the unique solution.
Practice: Augmented Matrix (~2016 Test2 #2)
Practice: Augmented Matrix
Answer the following questions about augmented matrices
Augmented Matrix
Answer the following questions about augmented matrices:
Practice: Augmented Matrix
Determine all values of a, ba,\ b and cc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r} 2&-1&0\\ 0&1&-3 \end{array}\right] represents the following system of linear equations:
ax1=bx2(a+c)x2+3=0\begin{array}{rcl} ax_1&=&bx_2\\ (a+c)x_2+3&=&0 \end{array}
Which of the following matrices is a result of performing a single elementary row operation on the augmented matrix [2−2051−43100−24]\left[\begin{array}{rrr|r} 2&-2&0&5\\ 1&-4&3&1\\ 0&0&-2&4 \end{array}\right]?
Consider a homogeneous system AX=0AX=0 where AA is a 2×42\times4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=0 is the unique solution.
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.
If * represents any arbitrary number, how many solutions does the linear system with the following augmented matrix have?
[010∗∗003∗∗000a0]\left[\begin{array}{cccc|c} 0 & 1 & 0 & * &* \\ 0 & 0 & 3 & * & *\\ 0 & 0 & 0 & a & 0 \end{array}\right]
Augmented matrices
If * represents any arbitrary number, how many solutions does the linear system with the following augmented matrix have?
[010∗∗003∗∗000a0]\left[\begin{array}{cccc|c} 0 & 1 & 0 & * &* \\ 0 & 0 & 3 & * & *\\ 0 & 0 & 0 & a & 0 \end{array}\right]
Linear equations and augmented matrices
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?
Linear equations and systems
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?
Homogenous system with linear equations
A homogeneous system (Ax⃗=0⃗A\vec{x}=\vec{0}) with mmlinear equations in nnunknowns has exactly one solution. Which of the following statements is always true?
Matrix problem
Given that A=[10−3725411]A=\begin{bmatrix}1 & 0 & -3\\7 & 2 & 5\\4 & 1 & 1\end{bmatrix}, which of the following statements is correct?
Augmented Matrix: Gauss-Jordan Elimination
Consider the system of linear equations
x2+4x3=14+2x1−4x4x1−x3−x4+3=02x4−6=0\begin{array}{c} x_2+4x_3=14+2x_1-4x_4\\ x_1-x_3-x_4+3=0\\ 2x_4-6=0 \end{array}
(a) Write the augmented matrix for the standard form of this system.