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Wize High School Algebra II Textbook (Common Core) > Systems of Linear Equations (Extension Topic)

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}