9. $1\times 1$

11. $2\times 4$

13. $2\times 3$

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

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

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

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

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

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

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

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

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

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

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

37. No, because Property 2 is not satisfied

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

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

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

45. $(5,-2);\{\begin{array}{rrrrr}x& +& 2y& =& 1\\ & & y& =& -2\end{array}$

47. $(-4-4y,y,3);\{\begin{array}{rrr}x+4y+2z& =& 2\\ z& =& 3\end{array}$

49. $(1,2,-1);\{\begin{array}{rrrrrrr}x& +& 2y& +& 3z& =& 2\\ & & y& -& 2z& =& 4\\ & & & & z& =& -1\end{array}$

51. $(2,-5,-2w+3,w);\{\begin{array}{rrrrrrrr}x& & & & & & =& 2\\ & & y& & & & =& -5\\ & & & z& +& 2w& =& 3\end{array}$

53. $(-5,4,3,0);\{\begin{array}{rrrrrrr}x& & & & & =& -5\\ & y& & & & =& 4\\ & & z& +& 2w& =& 3\\ & & & & w& =& 0\end{array}$

55. $\{(5,-3)\}$

57. $\{(3,1)\}$

59. $\left\{\left(-{\displaystyle \frac{1}{5}},-{\displaystyle \frac{23}{25}}\right)\right\}$

61. $\{(2,1)\}$

63. $\{(1,2,3)\}$

65. $\{(2,3,4)\}$

67. $\{(1,2,3)\}$

69. $\left\{\left({\displaystyle \frac{5}{2}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{2}}\right)\right\}$

71. $\{(1+4z,3-3z,z)\}$

73. $\{(1,2,-1)\}$

75.

1. One solution

2. Infinitely many solutions

3. No solution

77. False. Each row of A has seven entries.