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Solving a system of linear equations

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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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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.

(a) Write the augmented matrix of this linear system.

(b) Transform the augmented matrix into row echelon form using elementary row operations. Clearly indicate which operation you are performing at each step.

(c) Use your result from part (b) to find the solution set to the original system using back substitution.
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]
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}
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}
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.
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.