College Algebra, 4th Edition

Building Skills

1. 9. $1\times 1$

2. 11. $2\times 4$

3. 13. $2\times 3$

4. 15. $a_{13}=3,a_{31}=9,a_{33}=11,a_{34}=12$

5. 17. No, it is not a ­rectangular array of numbers.

6. 19. $\left[\begin{array}{rrr}2& 4& 2\\ 1& -3& 1\end{array}\right]$

7. 21. $\left[\begin{array}{rrr}5& -7& 11\\ -13& 17& 19\end{array}\right]$

8. 23. $\left[\begin{array}{rrrr}-1& 2& 3& 8\\ 2& -3& 9& 16\\ 4& -5& -6& 32\end{array}\right]$

9. 25. $\{\begin{array}{rrrrrrr}x& +& 2y& -& 3z& =& 4\\ -2x& -& 3y& +& z& =& 5\\ 3x& -& 3y& +& 2z& =& 7\end{array}$

10. 27. $\{\begin{array}{rrrrrrr}x& -& y& +& z& =& 2\\ 2x& +& y& -& 3z& =& 6\end{array}$

11. 29. $\left[\begin{array}{lll}1& 2& 3\\ 0& 1& 1\end{array}\right]$

12. 31. $\left[\begin{array}{rrrr}1& 2& 3& 4\\ 0& 1& 5& -3\\ 0& 0& 1& -1\end{array}\right]$

13. 33. $\begin{array}{lllllll}\left[\begin{array}{rrr}4& 5& -7\\ 5& 4& -2\end{array}\right]& \underrightarrow{{\displaystyle \frac{1}{4}}{R}_1}& \left[\begin{array}{rrr}1& {\displaystyle \frac{5}{4}}& -{\displaystyle \frac{7}{4}}\\ 5& 4& -2\end{array}\right]& \underrightarrow{-5R_1+R_2\to R_2}& \left[\begin{array}{rrr}1& {\displaystyle \frac{5}{4}}& -{\displaystyle \frac{7}{4}}\\ 0& -{\displaystyle \frac{9}{4}}& {\displaystyle \frac{27}{4}}\end{array}\right]& \underrightarrow{-{\displaystyle \frac{4}{9}}{R}_2}& \left[\begin{array}{rrr}1& {\displaystyle \frac{5}{4}}& -{\displaystyle \frac{7}{4}}\\ 0& 1& -3\end{array}\right]\end{array}$

14. 35. $\begin{array}{lllllll}\left[\begin{array}{rrrr}1& 4& 3& 1\\ 0& -3& -2& 0\\ 0& 7& 5& -3\end{array}\right]& \underrightarrow{-{\displaystyle \frac{1}{3}}{R}_2}& \left[\begin{array}{rrrr}1& 4& 3& 1\\ 0& 1& {\displaystyle \frac{2}{3}}& 0\\ 0& 7& 5& -3\end{array}\right]& \underrightarrow{-7R_2+R_3\to R_3}& \left[\begin{array}{rrrr}1& 4& 3& 1\\ 0& 1& {\displaystyle \frac{2}{3}}& 0\\ 0& 7& {\displaystyle \frac{1}{3}}& -3\end{array}\right]& \underrightarrow{3R_3}& \left[\begin{array}{rrrr}1& 4& 3& 1\\ 0& 1& {\displaystyle \frac{2}{3}}& 0\\ 0& 0& 1& -9\end{array}\right]\end{array}$

15. 37. No, because Property 2 is not satisfied

16. 39. The matrix is in reduced row-echelon form because Properties 1–4 are satisfied.

17. 41. The matrix is in reduced row-echelon form because Properties 1–4 are satisfied.

18. 43. The matrix is in reduced row-echelon form because Properties 1–4 are ...