Linear Algebra with Applications, 10th Edition

Section 1.2 Exercises

1. Which of the matrices that follow are in row echelon form? Which are in reduced row echelon form?

   1. $\left[\begin{array}{cccc}1& 2& 3& 4\\ 0& 0& 1& 2\end{array}\right]$

   2. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$

   3. $\left[\begin{array}{ccc}1& 3& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

   4. $\left[\begin{array}{cc}0& 1\\ 0& 0\\ 0& 0\end{array}\right]$

   5. $\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 0& 3\end{array}\right]$

   6. $\left[\begin{array}{ccc}1& 4& 6\\ 0& 0& 1\\ 0& 1& 3\end{array}\right]$

   7. $\left[\begin{array}{ccccc}1& 0& 0& 1& 2\\ 0& 1& 0& 2& 4\\ 0& 0& 1& 3& 6\end{array}\right]$

   8. $\left[\begin{array}{cccc}0& 1& 3& 4\\ 0& 0& 1& 3\\ 0& 0& 0& 0\end{array}\right]$

2. The augmented matrices that follow are in row echelon form. For each case, indicate whether the corresponding linear system is consistent. If the system has a unique solution, find it.

   1. $\left[\begin{array}{cc}1& 2\\ 0& 1\\ 0& 0\end{array}\right|\begin{array}{c}4\\ 3\\ 1\end{array}]$

   2. $\left[\begin{array}{cc}1& 3\\ 0& 1\\ 0& 0\end{array}\right|\begin{array}{r}1\\ -1\\ 0\end{array}]$

   3. $\left[\begin{array}{ccc}1& \hfill -2& \hfill 4\\ 0& \hfill 0& \hfill 1\\ 0& \hfill 0& \hfill 0\end{array}\right|\begin{array}{r}1\\ 3\\ 0\end{array}]$

   4. $\left[\begin{array}{ccc}1& \hfill -2& \hfill 2\\ 0& \hfill 1& \hfill -1\\ 0& \hfill 0& \hfill 1\end{array}\right|\begin{array}{r}-2\\ 3\\ 2\end{array}]$

   5. $\left[\begin{array}{ccc}1& \hfill 3& \hfill 2\\ 0& \hfill 0& \hfill 1\\ 0& \hfill 0& \hfill 0\end{array}\right|\begin{array}{r}-2\\ 4\\ 1\end{array}]$

   6. $\left[\begin{array}{ccc}\hfill 1& \hfill -1& \hfill 3\\ \hfill 0& \hfill 1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right|\begin{array}{c}\hfill 8\\ \hfill 7\\ \hfill 2\\ \hfill 0\end{array}]$

3. The augmented matrices that follow are in reduced row echelon form. In each case, find the solution set to the corresponding linear system.

   1. $\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right|\begin{array}{r}-2\\ 5\\ 3\end{array}]$

   2. $\left[\begin{array}{ccc}\hfill 1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right|\begin{array}{r}2\\ 3\\ 1\end{array}]$

   3. $\left[\begin{array}{ccc}\hfill 1& \hfill -3& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right|\begin{array}{r}2\\ -2\\ 0\end{array}]$

   4. $\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 0& 1& 3\end{array}\right|\begin{array}{c}5\\ 4\end{array}]$

   5. $[\begin{array}{cccc}1& 5& \hfill -2& 0\\ 0& 0& \hfill 0& 1\end{array}$