Augmented Matrix: Gauss-Jordan Elimination

Reducing a Matrix (Gauss-Jordan Elimination)

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

Augmented Matrix

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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}x2​+4x3​=14+2x1​−4x4​x1​−x3​−x4​+3=02x4​−6=0​

(a) Write the augmented matrix for the standard form of this system.

(b) Use your augmented matrix from part (a) to solve the system using Gauss-Jordan elimination.

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More Reducing a Matrix (Gauss-Jordan Elimination) Questions:

Gauss-Jordan Elimination $\tkct{LAT}$

This activity is just here as a convenient reminder / link for the Linear Algebra Toolkit. The solution illustrate a few points about how to use it; however, the actual details of row-reduction should be addressed in the context of the original chapter :)
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}}​036​−266​333​  1−2  5​​.

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}}​036​−266​333​  1−2  5​​.

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]​11−2​0−22​13−4​​110​​

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}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.

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}x1​+x2​2−x1​x1​+x2​+x3​​===​7−3x3​70​
2. Determine all values of a, ba,\ ba, b and ccc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r}
2&-1&0\\
0&1&-3
\end{array}\right][20​−11​0−3​] 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}ax1​(a+c)x2​+3​==​bx2​0​

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}​31−1​102​214​​

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]​132​3−61​−1−3−2​​

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]​132​3−61​−1−3−2​​
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]​624−2​−18−6−126​12484​3000​02−10​​ into row-reduce echelon form (RREF) and then state the solution.

Let AAA 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]​1−23​202​01−1​​.
Find the first row of AAA.

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]​210​−2−40​03−2​514​​?

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}−2x1​−4x2​−6x3​−19x4​3x1​+10x2​−30x3​+9x4​x1​+3x2​−6x3​+4x4​7x1​+17x2​+45x4​​====​−1524855​
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}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
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}−2x1​−4x2​−6x3​−19x4​3x1​+10x2​−30x3​+9x4​x1​+3x2​−6x3​+4x4​7x1​+17x2​+45x4​​====​−1524855​
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}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
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}−2x1​−4x2​−6x3​−19x4​3x1​+10x2​−30x3​+9x4​x1​+3x2​−6x3​+4x4​7x1​+17x2​+45x4​​====​−1524855​
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4)(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}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
where we wish to solve for the triple (x1,x2,x3)(x_1,x_2,x_3)(x1​,x2​,x3​) of real numbers that satisfy this system

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}−x1​+2x3​5x1​+x2​10x3​+2x2​​===​5−x2​9+x3​1−x1​​

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}x1​+6x2​−x3​−5x4​x1​−x2​+4x4​−3x2​−3x3​+x4​2x1​+3x2​−2x3​+x4​​====​360−1​

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}x1​+x2​2−x1​x1​+x2​+x3​​===​7−3x3​70​

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}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
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}x−y=5z+72x​−z=30=4−x−8y​
                                                

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}}⎩⎨⎧​x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6​​				
a) Construct the augmented matrix A\bcb{A}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]
​102​120​−35−4​​630​​

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}x1​+x2​2−x1​x1​+x2​+x3​​===​7−3x3​70​
2. Determine all values of a, ba,\ ba, b and ccc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r}
2&-1&0\\
0&1&-3
\end{array}\right][20​−11​0−3​] 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}ax1​(a+c)x2​+3​==​bx2​0​

Consider a homogeneous system AX=0AX=0AX=0 where AAA is a 2×42\times42×4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=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,\ ba, b and ccc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r}
2&-1&0\\
0&1&-3
\end{array}\right][20​−11​0−3​] 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}ax1​(a+c)x2​+3​==​bx2​0​

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]​210​−2−40​03−2​514​​?

Consider a homogeneous system AX=0AX=0AX=0 where AAA is a 2×42\times42×4 matrix. Consider the following statements:
(i) The system is inconsistent.
(ii) X=0X=0X=0 is the unique solution.

Find all the values of kkk 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]​100​210​00k2−9​12k−3​​has infinitely many solutions.

Find all the values of kkk 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]​100​210​00k2−9​12k−3​​ 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]​000​100​030​∗∗a​∗∗0​​

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]​000​100​030​∗∗a​∗∗0​​

Linear equations and augmented matrices

Suppose that a system of mmmlinear equations in nnnvariables is represented by the augmented matrix [A∣b][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 AAAis 3, which of the following statements best describes the system?

Homogenous system with linear equations

A homogeneous system (Ax⃗=0⃗A\vec{x}=\vec{0}Ax=0) with mmmlinear equations in nnnunknowns 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}A=​174​021​−351​​, which of the following statements is correct?

Augmented Matrix: Gauss-Jordan Elimination

Related Topics

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Gauss-Jordan Elimination $\tkct{LAT}$

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More Augmented Matrix Questions:

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Writing the augmented matrix of a system of linear equations

Practice: Augmented Matrix

Solving a system of linear equations

Augmented Matrix

123 - FML #1 - 18.1W - e.g. 20

Additional Practice Problems--Systems of Linear Equations

Practice: Augmented Matrix (~2016 Test2 #2)

Augmented Matrix

Practice: Augmented Matrix

Augmented matrices

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