Difference between revisions of "Science:Math Exam Resources/Courses/MATH152/April 2015/Question B 1 (c)/Solution 1"

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We operate Gaussian elimination on the rows and find row echelon form;
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We perform Gaussian elimination to compute the reduced row echelon form of the augmented matrix:
  
<math>
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:<math>
 
\begin{align}
 
\begin{align}
\begin{pmatrix}1&1&1&600\\-1&1&0&200\\1&1&-1&0 \end{pmatrix}&\xrightarrow{R_3-R_1}
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\left[\begin{array}{ccc|c}1&1&1&600\\-1&1&0&200\\1&1&-1&0 \end{array}\right]
\begin{pmatrix}1&1&1&600\\-1&1&0&200\\0&0&-2&-600 \end{pmatrix}\xrightarrow{R_2+R_1}
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&\xrightarrow{R_3-R_1} \left[\begin{array}{ccc|c}1&1&1&600\\-1&1&0&200\\0&0&-2&-600 \end{array}\right]
\begin{pmatrix}1&1&1&600\\0&2&1&800\\0&0&-2&-600 \end{pmatrix}\xrightarrow{\frac{R_2}2, -\frac{R_3}2}
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&\xrightarrow{R_2+R_1} \left[\begin{array}{ccc|c}1&1&1&600\\0&2&1&800\\0&0&-2&-600 \end{array}\right]
\begin{pmatrix}1&1&1&600\\0&1&1/2&400\\0&0&1&300 \end{pmatrix}\\
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&\xrightarrow{\frac{R_2}2, -\frac{R_3}2} \left[\begin{array}{ccc|c}1&1&1&600\\0&1&1/2&400\\0&0&1&300 \end{array}\right] \\
&\xrightarrow{R_1-R_2}  
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&\xrightarrow{R_1-R_2} \left[\begin{array}{ccc|c}1&0&1/2&200\\0&1&1/2&400\\0&0&1&300 \end{array}\right]
\begin{pmatrix}1&0&1/2&200\\0&1&1/2&400\\0&0&1&300 \end{pmatrix}\xrightarrow{R_2-\frac 12 R_3}
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&\xrightarrow{R_2-\frac 12 R_3} \left[\begin{array}{ccc|c}1&0&1/2&200\\0&1&0&250\\0&0&1&300 \end{array}\right]
\begin{pmatrix}1&0&1/2&200\\0&1&0&250\\0&0&1&300 \end{pmatrix}\xrightarrow{R_1-\frac 12 R_3}
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&\xrightarrow{R_1-\frac 12 R_3} \left[\begin{array}{ccc|c}1&0&0&50\\0&1&0&250\\0&0&1&300 \end{array}\right].
\begin{pmatrix}1&0&0&50\\0&1&0&250\\0&0&1&300 \end{pmatrix}.\end{align}</math>  
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\end{align}
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</math>  
  
Now it becomes  <math display="block">\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}50\\250\\300\end{pmatrix}.</math>
 
 
Thus <math display="inline">\color{blue}x_1=50, x_2=250, x_3=300.</math>
 
Thus <math display="inline">\color{blue}x_1=50, x_2=250, x_3=300.</math>

Latest revision as of 20:26, 8 March 2018

We perform Gaussian elimination to compute the reduced row echelon form of the augmented matrix:

Thus