1.3.1: Exercises 1.3

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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.

Exercise \(\PageIndex{1}\)

\(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{1}\\{1}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{1}\\{0}&{1}&{2}\end{array}\right]\)

Answer

Exercise \(\PageIndex{2}\)

\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{1}\\{0}&{1}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{0}&{0}\\{1}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)

Answer

Exercise \(\PageIndex{3}\)

\(\left[\begin{array}{ccc}{1}&{1}&{1}\\{0}&{1}&{1}\\{0}&{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{1}\\{0}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{-5}\\{0}&{1}&{0}&{7}\\{0}&{0}&{1}&{3}\end{array}\right]\)

Answer

Exercise \(\PageIndex{4}\)

\(\left[\begin{array}{cccc}{2}&{0}&{0}&{2}\\{0}&{2}&{0}&{2}\\{0}&{0}&{2}&{2}\end{array}\right]\)
\(\left[\begin{array}{cccc}{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{cccc}{0}&{0}&{1}&{-5}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{cccccc}{1}&{1}&{0}&{0}&{1}&{1}\\{0}&{0}&{1}&{0}&{1}&{1}\\{0}&{0}&{0}&{1}&{0}&{0}\end{array}\right]\)

Answer

In Exercises \(\PageIndex{5}\) - \(\PageIndex{22}\), use Gaussian Elimination to put the given matrix into reduced row echelon form.

Exercise \(\PageIndex{5}\)

\(\left[\begin{array}{cc}{1}&{2}\\{-3}&{-5}\end{array}\right]\)

Answer: \(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)

Exercise \(\PageIndex{6}\)

\(\left[\begin{array}{cc} 2&-2\\3&-2\end{array}\right]\)

Answer: \(\left[\begin{array}{cc} 1&0\\0&1\end{array}\right]\)

Exercise \(\PageIndex{7}\)

\(\left[\begin{array}{cc} 4&12\\-2&-6\end{array}\right]\)

Answer: \(\left[\begin{array}{cc} 1&3\\0&0\end{array}\right]\)

Exercise \(\PageIndex{8}\)

\(\left[\begin{array}{cc} -5&7\\10&14\end{array}\right]\)

Answer: \(\left[\begin{array}{cc} 1&-7/5\\0&0\end{array}\right]\)

Exercise \(\PageIndex{9}\)

\(\left[\begin{array}{ccc} -1&1&4\\-2&1&1\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&3\\0&1&7\end{array}\right]\)

Exercise \(\PageIndex{10}\)

\(\left[\begin{array}{ccc} 7&2&3\\3&1&2\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&-1\\0&1&5\end{array}\right]\)

Exercise \(\PageIndex{11}\)

\(\left[\begin{array}{ccc} 3&-3&6\\-1&1&-2\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&-1&2\\0&0&0\end{array}\right]\)

Exercise \(\PageIndex{12}\)

\(\left[\begin{array}{ccc} 4&5&-6\\-12&-15&18\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&\frac54&-\frac32\\0&0&0\end{array}\right]\)

Exercise \(\PageIndex{13}\)

\(\left[\begin{array}{ccc} -2&-4&-8\\-2&-3&-5\\ 2&3&6\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)

Exercise \(\PageIndex{14}\)

\(\left[\begin{array}{ccc} 2&1&1\\1&1&1\\2&1&2\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)

Exercise \(\PageIndex{15}\)

\(\left[\begin{array}{ccc} 1&2&1\\1&3&1\\-1&-3&0\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)

Exercise \(\PageIndex{16}\)

\(\left[\begin{array}{ccc} 1&2&3\\0&4&5\\1&6&9\end{array}\right]\)

Answer: \(\left[\begin{array}{ccc} 1&0&0\\0&1&0\\0&0&1\end{array}\right]\)

Exercise \(\PageIndex{17}\)

\(\left[\begin{array}{cccc} 1&1&1&2\\2&-1&-1&1\\-1&1&1&0\end{array}\right]\)

Answer: \(\left[\begin{array}{cccc} 1&0&0&1\\0&1&1&1\\0&0&0&0\end{array}\right]\)

Exercise \(\PageIndex{18}\)

\(\left[\begin{array}{cccc} 2&-1&1&5\\3&1&6&-1\\3&0&5&0\end{array}\right]\)

Answer: \(\left[\begin{array}{cccc} 1&0&0&5\\0&1&0&2\\0&0&1&-3\end{array}\right]\)

Exercise \(\PageIndex{19}\)

\(\left[\begin{array}{cccc} 1&1&-1&7\\2&1&0&10\\3&2&-1&17\end{array}\right]\)

Answer: \(\left[\begin{array}{cccc} 1&0&1&3\\0&1&-2&4\\0&0&0&0\end{array}\right]\)

Exercise \(\PageIndex{20}\)

\(\left[\begin{array}{cccc} 4&1&8&15\\1&1&2&7\\3&1&5&11\end{array}\right]\)

Answer: \(\left[\begin{array}{cccc} 1&0&3&4\\0&1&-1&3\\0&0&0&0\end{array}\right]\)

Exercise \(\PageIndex{21}\)

\(\left[\begin{array}{cccccc} 2&2&1&3&1&4\\1&1&1&3&1&4\end{array}\right]\)

Answer: \(\left[\begin{array}{cccccc} 1&1&0&0&0&0\\0&0&1&3&1&4\end{array}\right]\)

Exercise \(\PageIndex{22}\)

\(\left[\begin{array}{cccccc} 1&-1&3&1&-2&9\\2&-2&6&1&-2&13\end{array}\right]\)

Answer: \(\left[\begin{array}{cccccc} 1&-1&3&0&0&4\\0&0&0&1&-2&5\end{array}\right]\)