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

Augmented Matrix

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Augmented Matrix

Matrices

  • A matrix is a rectangular array of numbers:
[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]\left[\begin{array}{cccc} a_{11} & a_{12} &\cdots & a_{1n}\\ a_{21} & a_{22} &\cdots & a_{2n}\\ \vdots &\vdots & ⋱ & \vdots\\ a_{m1} & a_{m2} &\cdots & a_{mn}\\ \end{array}\right]
  • The size of the matrix is written as m×nm\times n where:
  • dimension mm is the number of rows (horizontal)
  • dimension nn is the number of columns (vertical)
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Augmented Matrices

A linear system can be represented by an augmented matrix.
The entries in the augmented matrix are the same aij\colorThree{a_{ij}} and bi\colorTwo{b_i} that appear in the SLE:
m equations {a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮am1x1+am2x2+⋯+amnxn=bmm \text{ equations } \left\{ \begin{array}{ccccccccl} \colorThree{a_{11}}{{x_1}}&+&\colorThree{a_{12}}{{x_2}}&+&\cdots&+&\colorThree{a_{1n}}{{x_n}} &=& \colorTwo{b_1}\\ \colorThree{a_{21}}{x_1}&+&\colorThree{a_{22}}{x_2}&+&\cdots&+&\colorThree{a_{2n}}{x_n} &=& \colorTwo{b_2}\\ &&&\vdots&&\\ \colorThree{a_{m1}}{x_1}&+&\colorThree{a_{m2}}{x_2}&+&\cdots&+&\colorThree{a_{mn}}{x_n} &=& \colorTwo{b_m}\\ \end{array} \right.
The augmented matrix consists of a coefficient matrix on the left, and the augmented column (or constant vector) on the right:
[a11a12⋯a1nb1a21a22⋯a2nb2⋮⋮⋱⋮⋮am1am2⋯amnbm]\left[\begin{array}{cccc|c} \colorThree{a_{11}} & \colorThree{a_{12}} &\cdots & \colorThree{a_{1n}} & \colorTwo{b_1} \\ \colorThree{a_{21}} & \colorThree{a_{22}} &\cdots & \colorThree{a_{2n}} & \colorTwo{b_2} \\ \vdots &\vdots & ⋱ & \vdots &\vdots \\ \colorThree{a_{m1}} & \colorThree{a_{m2}} &\cdots & \colorThree{a_{mn}} & \colorTwo{b_m} \\ \end{array}\right]
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Example: Augmented Matrix

Convert the following SLE into an augmented matrix.
State the dimensions of the coefficient matrix.
x−y=5z+7x2−z=30=4−x−π2y\begin{array}{rcl} x-y&=&5z+7\\[0.3em] \frac{x}{2}-z&=&3\\[0.3em] 0&=&4-x-\pi^2y \end{array}
Rewriting in standard form:
1x−1y−5z=712x+0y−1z=31x+π2y+0z=4\begin{array}{rcl} 1x&-1y&-5z&=&7\\ \frac{1}{2}x&+0y &-1z&=&3\\ 1x&+\pi^2y&+0z &=&4 \end{array}
Writing the coefficients on the left and the constants on the right:

[1−1−5120−11π20∣734]\left[\begin{array}{l} 1 & -1 & -5 \\ \frac{1}{2} &0 & -1 \\ 1 & \pi^2 & 0 \\ \end{array}\right.\left|\begin{array}{l} 7\\3\\4\end{array}\right]

The coefficient matrix (on the left) has 3 rows and 3 columns.
Therefore, the dimensions of the coefficient matrix are 3×33 \times 3.
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Example: Augmented Matrix

Write the augmented matrix of the following system of linear equations:
2x1−x2+3x3+4x4=9x1−2x3+7x4=113x1−3x2+x3+5x4=810x4+2x1+x2+3x3=21\begin{array}{rcr} 2x_1-x_2+3x_3+4x_4&=&9\\[0.5em] x_1 -2x_3+7x_4&=& 11\\[0.5em] 3x_1-3x_2+x_3+5x_4&=&8\\[0.5em] 10x_4+2x_1+x_2+3x_3&=&21 \end{array}
Write the unknowns in the same order in each equation: x1x2x3x4x_1\quad x_2\quad x_3\quad x_4
Write coefficients in front of every variable (including 1s and 0s)
The system is rewritten as:
2x1−1x2+3x3+4x4=91x1+0x2−2x3+7x4=113x1−3x2+1x3+5x4=82x1+1x2+3x3+10x4=21\begin{array}{rcrcrcrcr} 2x_1&-&1x_2&+&3x_3&+&4x_4&=&9\\[0.5em] 1x_1 &+&0x_2&-&2x_3&+&7x_4&=& 11\\[0.5em] 3x_1&-&3x_2&+&1x_3&+&5x_4&=&8\\[0.5em] 2x_1&+&1x_2&+&3x_3&+&10x_4&=&21 \end{array}

To write the augmented matrix, write the coefficients in a rectangular array:
[2−134910−27113−31582131021]\left[\begin{array}{rrrr|r} 2&-1&3&4&9\\ 1&0&-2&7& 11\\ 3&-3&1&5&8\\ 2&1&3&10&21 \end{array}\right]

Given the augmented matrix [2−1001−3]\left[\begin{array}{rr|r} 2&-1&0\\ 0&1&-3 \end{array}\right]
determine the values of a, ba,\ b and cc if the augmented matrix 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}
Answer the following questions about augmented matrices:
What is the augmented matrix that represents this system of linear equations? [Input the entries in the 2×42 \times 4 table]

3x−y2=3−0.5z0=7−y+3z\begin{array}{rcl} 3x-\dfrac{y}{2}&=&3-0.5z\\[1em] 0&=&7-y+3z \end{array}

Extra Practice
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.
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.