11.5.1: Elementary Row Operations and Gaussian Elimination (Exercises)
- Page ID
- 120066
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In Exercises \(\PageIndex{1}\) - \(\PageIndex{4}\), state whether or not the given matrices are in reduced row echelon form. If it is not, state why.
- \(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)
- \(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)
- \(\left[\begin{array}{cc}{1}&{1}\\{1}&{1}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{1}&{0}&{1}\\{0}&{1}&{2}\end{array}\right]\)
- Answer
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- yes
- no
- no
- yes
- \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{1}&{0}&{1}\\{0}&{1}&{1}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{0}&{0}&{0}\\{1}&{0}&{0}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)
- Answer
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- yes
- yes
- no
- yes
- \(\left[\begin{array}{ccc}{1}&{1}&{1}\\{0}&{1}&{1}\\{0}&{0}&{1}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{0}\end{array}\right]\)
- \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{1}\\{0}&{0}&{0}\end{array}\right]\)
- \(\left[\begin{array}{cccc}{1}&{0}&{0}&{-5}\\{0}&{1}&{0}&{7}\\{0}&{0}&{1}&{3}\end{array}\right]\)
- Answer
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- no
- yes
- yes
- yes
- \(\left[\begin{array}{cccc}{2}&{0}&{0}&{2}\\{0}&{2}&{0}&{2}\\{0}&{0}&{2}&{2}\end{array}\right]\)
- \(\left[\begin{array}{cccc}{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}\end{array}\right]\)
- \(\left[\begin{array}{cccc}{0}&{0}&{1}&{-5}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\end{array}\right]\)
- \(\left[\begin{array}{cccccc}{1}&{1}&{0}&{0}&{1}&{1}\\{0}&{0}&{1}&{0}&{1}&{1}\\{0}&{0}&{0}&{1}&{0}&{0}\end{array}\right]\)
- Answer
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- no
- yes
- yes
- yes
In Exercises \(\PageIndex{5}\) - \(\PageIndex{22}\), use Gaussian Elimination to put the given matrix into reduced row echelon form.
\(\left[\begin{array}{cc}{1}&{2}\\{-3}&{-5}\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc} 2&-2\\3&-2\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cc} 1&0\\0&1\end{array}\right]\)
\(\left[\begin{array}{cc} 4&12\\-2&-6\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cc} 1&3\\0&0\end{array}\right]\)
\(\left[\begin{array}{cc} -5&7\\10&14\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cc} 1&-7/5\\0&0\end{array}\right]\)
\(\left[\begin{array}{ccc} -1&1&4\\-2&1&1\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&3\\0&1&7\end{array}\right]\)
\(\left[\begin{array}{ccc} 7&2&3\\3&1&2\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&-1\\0&1&5\end{array}\right]\)
\(\left[\begin{array}{ccc} 3&-3&6\\-1&1&-2\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&-1&2\\0&0&0\end{array}\right]\)
\(\left[\begin{array}{ccc} 4&5&-6\\-12&-15&18\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&\frac54&-\frac32\\0&0&0\end{array}\right]\)
\(\left[\begin{array}{ccc} -2&-4&-8\\-2&-3&-5\\ 2&3&6\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)
\(\left[\begin{array}{ccc} 2&1&1\\1&1&1\\2&1&2\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)
\(\left[\begin{array}{ccc} 1&2&1\\1&3&1\\-1&-3&0\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)
\(\left[\begin{array}{ccc} 1&2&3\\0&4&5\\1&6&9\end{array}\right]\)
- Answer
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\(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)
\(\left[\begin{array}{cccc} 1&1&1&2\\2&-1&-1&1\\-1&1&1&0\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccc} 1&0&0&1\\0&1&1&1\\0&0&0&0\end{array}\right]\)
\(\left[\begin{array}{cccc} 2&-1&1&5\\3&1&6&-1\\3&0&5&0\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccc} 1&0&0&5\\0&1&0&2\\0&0&1&-3\end{array}\right]\)
\(\left[\begin{array}{cccc} 1&1&-1&7\\2&1&0&10\\3&2&-1&17\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccc} 1&0&1&3\\0&1&-2&4\\0&0&0&0\end{array}\right]\)
\(\left[\begin{array}{cccc} 4&1&8&15\\1&1&2&7\\3&1&5&11\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccc} 1&0&3&4\\0&1&-1&3\\0&0&0&0\end{array}\right]\)
\(\left[\begin{array}{cccccc} 2&2&1&3&1&4\\1&1&1&3&1&4\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccccc} 1&1&0&0&0&0\\0&0&1&3&1&4\end{array}\right]\)
\(\left[\begin{array}{cccccc} 1&-1&3&1&-2&9\\2&-2&6&1&-2&13\end{array}\right]\)
- Answer
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\(\left[\begin{array}{cccccc} 1&-1&3&0&0&4\\0&0&0&1&-2&5\end{array}\right]\)
In Exercises \(\PageIndex{23}\) - \(\PageIndex{36}\), find the solution to the given linear system. If the system has infinite solution, give 2 solutions.
\(\begin{array}{ccccc} 2x_1&+&4x_2&=&2\\ x_1&+&2x_2&=&1\\ \end{array}\)
- Answer
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\(x_1=1-2x_2\); \(x_2\) is free. Possible solutions: \(x_1=1\), \(x_2=0\) and \(x_1=-1\), \(x_2=1\).
\(\begin{array}{ccccc} -x_1&+&5x_2&=&3\\ 2x_1&-&10x_2&=&-6\\ \end{array}\)
- Answer
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\(x_1=-3+5x_2\); \(x_2\) is free. Possible solutions: \(x_1 = 3\), \(x_2=0\) and \(x_1 = -8\), \(x_2 = -1\)
\(\begin{array}{ccccc} x_1&+&x_2&=&3\\ 2x_1&+&x_2&=&4\\ \end{array}\)
- Answer
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\(x_1=1\); \(x_2=2\)
\(\begin{array}{ccccc} -3x_1&+&7x_2&=&-7\\ 2x_1&-&8x_2&=&8\\ \end{array}\)
- Answer
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\(x_1=0\); \(x_2=-1\)
\(\begin{array}{ccccc} 2x_1&+&3x_2&=&1\\ -2x_1&-&3x_2&=&1\\ \end{array}\)
- Answer
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No solution; the system is inconsistent.
\(\begin{array}{ccccc} x_1&+&2x_2&=&1\\ -x_1&-&2x_2&=&5\\ \end{array}\)
- Answer
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No solution; the system is inconsistent.
\(\begin{array}{ccccccc} -2x_1&+&4x_2&+&4x_3&=&6\\ x_1&-&3x_2&+&2x_3&=&1\\ \end{array}\)
- Answer
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\(x_1=-11+10x_3\); \(x_2=-4+4x_3\); \(x_3\) is free. Possible solutions: \(x_1=-11\), \(x_2 = -4\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\) and \(x_3 = 1\).
\(\begin{array}{ccccccc} -x_1&+&2x_2&+&2x_3&=&2\\ 2x_1&+&5x_2&+&x_3&=&2\\ \end{array}\)
- Answer
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\(x_1=-\frac23+\frac89x_3\); \(x_2=\frac23-\frac59x_3\); \(x_3\) is free. Possible solutions: \(x_1 = -\frac23\), \(x_2 = \frac23\), \(x_3 = 0\) and \(x_1 = \frac49\), \(x_2 = -\frac19\), \(x_3 = 1\)
\(\begin{array}{rcl} -x_1-x_2+x_3+x_4&=&0\\ -2x_1-2x_2+x_3&=&-1\\ \end{array}\)
- Answer
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\(x_1=1-x_2-x_4\); \(x_2\) is free; \(x_3=1-2x_4\); \(x_4\) is free. Possible solutions: \(x_1 = 1\), \(x_2 = 0\), \(x_3 = 1\), \(x_4 = 0\) and \(x_1 = -2\), \(x_2 = 1\), \(x_3 = -3\), \(x_4=2\)
\(\begin{array}{rcl} x_1+x_2+6x_3+9x_4&=&0\\ -x_1-x_3-2x_4&=&-3\\ \end{array}\)
- Answer
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\(x_1=3-x_3-2x_4\); \(x_2=-3-5x_3-7x_4\); \(x_3\) is free; \(x_4\) is free. Possible solutions: \(x_1 =3\), \(x_2 = -3\), \(x_3=0\), \(x_4=0\) and \(x_1 = 0\), \(x_2 = -5\), \(x_3 =-1\), \(x_4=1\)
\(\begin{array}{ccccccc} 2x_1&+&x_2&+&2x_3&=&0\\ x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&2x_2&+&5x_3&=&3\\ \end{array}\)
- Answer
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No solution; the system is inconsistent.
\(\begin{array}{ccccccc} x_1&+&3x_2&+&3x_3&=&1\\ 2x_1&-&x_2&+&2x_3&=&-1\\ 4x_1&+&5x_2&+&8x_3&=&2\\ \end{array}\)
- Answer
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No solution; the system is inconsistent.
\(\begin{array}{ccccccc} x_1&+&2x_2&+&2x_3&=&1\\ 2x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&3x_2&+&5x_3&=&2\\ \end{array}\)
- Answer
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\(x_1=\frac13-\frac43x_3\); \(x_2=\frac13-\frac13x_3\); \(x_3\) is free. Possible solutions: \(x_1 = \frac13\), \(x_2=\frac13\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\), \(x_3=1\)
\(\begin{array}{ccccccc} 2x_1&+&4x_2&+&6x_3&=&2\\ 1x_1&+&2x_2&+&3x_3&=&1\\ -3x_1&-&6x_2&-&9x_3&=&-3\\ \end{array}\)
- Answer
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\(x_1=1-2x_2-3x_3\); \(x_2\) is free; \(x_3\) is free. Possible solutions: \(x_1=1\), \(x_2=0\), \(x_3=0\) and \(x_1=8\), \(x_2=1\), \(x_3 = -3\)
In Exercises \(\PageIndex{37}\) - \(\PageIndex{40}\), state for which values of \(k\) the given system will have exactly 1 solution, infinite solutions, or no solution.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\2x_1&+&4x_2&=&k\end{array}\)
- Answer
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Never exactly 1 solution; infinite solutions if \(k=2\); no solution if \(k\neq 2\).
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&1\end{array}\)
- Answer
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Exactly 1 solution if \(k\neq 2\); infinite solutions if \(k=2\); never no solution.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&2\end{array}\)
- Answer
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Exactly 1 solution if \(k\neq 2\); no solution if \(k=2\); never infinite solutions.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&3x_2&=&k\end{array}\)
- Answer
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Exactly 1 solution for all \(k\).