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Additional Practice Problems--Systems of Linear Equations

Related Topics

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

Reducing a Matrix (Gauss-Jordan Elimination)

6 Activities

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

Augmented Matrix

5 Activities

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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}

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3. Given the augmented matrix [23−1104∣−1−12]\left[\begin{array}{l} 2 &3 \\ -1 & 1 \\ 0&4 \end{array}\right.\left|\begin{array}{l} -1\\-1\\2\end{array}\right] , perform the following sequence of EROs:
i. Switch row 1 and row 2
ii. Multiply row 1 by negative one
iii. Add negative two times row 1 to row 2
iv. Multiply row 2 by 15\frac{1}{5}
v. Add row 2 to row 1 and add negative four times row 2 to row 3

4.) For what value(s) of kk will the following equation be linear in x, y, zx,\ y,\ z and a, b, ca,\ b,\ c?
2x+kay−(ln⁡2)b=3−c2x+\frac{k}{ay}-\left(\ln2\right)b=3-c
More Reducing a Matrix (Gauss-Jordan Elimination) Questions:
Gauss-Jordan Elimination $\tkct{LAT}$
Gauss-Jordan Elimination $\tkct{LAT}$
Use Gauss-Jordan Elimination to find the RREF of the matrix [0−23  1363−2663  5]\bcb{\begin{bmatrix} 0&-2&3&\ \ 1\\3&6&3&-2\\ 6&6&3&\ \ 5 \end{bmatrix}}.
Given the following augmented matrix of a system of equations, find the reduced row echelon form:
[1011−23−22−4∣110]\left[\begin{array}{l} 1&0&1\\ 1&-2&3\\ -2&2&-4\\ \end{array}\right.\left|\begin{array}{l}1 \\1\\0\end{array}\right]
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.
Reducing a Matrix
Practice: Reducing a Matrix
Find the reduced row echelon form (RREF) of the matrix:
[312101−124]\begin{bmatrix} 3&1&2\\ 1&0&1\\ -1&2&4 \end{bmatrix}
Reducing a Matrix
Practice: Reducing a Matrix
Given the matrix,
[13−13−6−321−2]\left[\begin{array}{ccc} 1&3&-1\\ 3&-6&-3\\ 2&1&-2 \end{array}\right]
Reducing a Matrix
Given the matrix,
[13−13−6−321−2]\left[\begin{array}{ccc} 1&3&-1\\ 3&-6&-3\\ 2&1&-2 \end{array}\right]
find the first row of the RREF of this matrix.
Practice Question: Matrix Reduction
Practice Question: Matrix Reduction
Reduce [6−1812302−64024−1280−1−26400]\left[\begin{array}{rrrr|r} 6 & -18 & 12 &3 &0\\ 2 &-6 & 4 &0&2\\ 4 & -12 & 8 &0&-1\\ -2&6&4&0&0 \end{array}\right] into row-reduce echelon form (RREF) and then state the solution.
Let AA be the row-reduced echelon form of the matrix [120−20132−1]\left[\begin{array}{r} 1&2&0\\ -2&0&1\\ 3&2&-1 \end{array}\right].
Find the first row of AA.
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]?
Solving a system of linear equations
Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1, x2, x3, x4) of real numbers that satisfy this system.
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.
Solving a system of linear equations
Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1, x2, x3, x4) of real numbers that satisfy this system.
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.
Solving a system of linear equations
Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) of real numbers that satisfy this system
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)(x_1,x_2,x_3) of real numbers that satisfy this system
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.
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]
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.
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.
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.