16.10 答复-第10章:系统和矩阵

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  答复----第10章:系统和矩阵

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  Section 10.2: Systems of Two Equations and Two Unknowns
  ::第10.2节:两个等同和两个未知的系统

  1. (-7, 3)
  2. (3, 8)
  3. $\begin{array}{r}\left(\frac{143}{18},\frac{\text{-}28}{9}\right)\end{array}$  
  4. (6, 10)
  5. (3, 9)
  6. (8, 2)
  7. (10, -10)
  8. (8, -5)
  9. (0, 5)
  10. $\begin{array}{r}\left(\frac{33}{19},\frac{3}{19}\right)\end{array}$  
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  11. A system of equations has no solution if, when you are solving, you end up with a false statement like 0 = 2. 
      ::等式系统无法解决问题,如果在解决时,最后出现0=2的虚假声明。
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  12. Answers will vary. Possible answers: If a system of equations has no solution, it means that the graphs do not intersect. For a system of two variables , the lines would be parallel.
      ::答案会有所不同。 可能的答案是: 如果一个方程式系统没有解答, 这意味着图形不会交叉。 对于由两个变量组成的系统, 线条将是平行的 。
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  13. A system where the two equations are equivalent (one is a multiple of the other) will produce an infinite number of solutions.
      ::两个方程式等同的系统(一个是另一个的倍数)将产生无限数量的解决方案。
  14. (2, 9)
  15. (7, 1)
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  16. 99 and 51
      ::99和51
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  17. $68,750 in Company A, and $31,250 in Company B
      ::A公司68 750美元,B公司31 250美元
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  18. Clownfish cost $1.95, and goldfish cost $1.25.
      ::小丑鱼费用为1.95美元,金鱼费用为1.25美元。
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  19. 26 and 9
      ::第26和9条
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  20. 5 cappuccinos and 4 lattes
      ::5个卡布奇诺和4个拿铁

   

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  Section 10.3: Solving Linear Systems in Three Variables
  ::第10.3节:三个变量中的解决线性系统

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  1. Yes
     ::是 是
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  2. No
     ::否 无
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  3. Yes
     ::是 是
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  4. Yes
     ::是 是
  5. (2, 7, 8)
  6. (-4, 1, 6)
  7. (5, 9, -2)
  8. (2, 1, 2)
  9. (- $\begin{array}{r}\frac{8}{3}\end{array}$ , 0, 0)
  10. (-7, 5, 4)
  11. (1, -4, 0)
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  12. No solution
      ::无解决方案
  13. (-1, 0, -1)
  14. (4, -2, 1)
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  15. $\begin{array}{r}y=3-x;z=0\end{array}$  
      ::y=3 - x; z=0
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  16.  
      $\begin{array}{rl}A+O+B& =9\\ 2A+2O& =10\\ 2O+B& =10\end{array}$  
      Apples cost $2; onions cost $3; a basket of blueberries costs $4.
      ::A+O+B=92A+2O=102O+B=10苹果费用2美元;洋葱费用3美元;蓝莓篮子费用4美元。
  17. 735
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  18. $24,000 in the savings account; $11,000 in the time deposit; $7,000 in the bond
      ::储蓄账户24 000美元;定期存款11 000美元;保证金7 000美元
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  19. 7 nickels, 6 dimes, and 12 quarters
      ::7个镍、6个零和12个季度
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  20. 126 adult, 92 children, and 42 student tickets
      ::126名成人、92名儿童和42张学生票

   

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  Section 10.4: Matrices to Represent Data
  ::第10.4节:将数据表示数

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  1. 2 x 4
     ::2 x 4
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  2. 2 x 2
     ::2 x 2
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  3. 4 x 2
     ::4 x 2 4 x 2
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  4. 4 x 3
     ::4 x 3 4 x 3
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  5. 1 x 2
     ::1 x 2
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  6. Answers will vary.
     ::答案将各有不同。
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  7. Answers will vary.
     ::答案将各有不同。
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  8. A diagonal matrix
     ::对角矩阵
  9. $\begin{array}{r}\left[\begin{array}{ccc}0& \text{-}1& \text{-}2\\ 1& 0& \text{-}1\end{array}\right]\end{array}$  
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  10. The rows represent the days of the week, and the columns represent the weeks of working.
      ::各行代表每周的天数,各栏代表工作周。
  11. $124
  12. $83
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  13. Fridays
      ::星期五 星期五
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  14. Thursdays
      ::星期四 星期四
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  15. No. The entries of matrices must be numbers, not words.
      ::否。矩阵条目必须是数字,而不是单词。

   

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  Section 10.5: Matrix Algebra
  ::第10.5节:矩阵体代数

  1.   $\begin{array}{r}\left[\begin{array}{cc}35& 26\\ 50& 34\end{array}\right]\end{array}$  
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  2. Not possible—you can't multiply a 2 x 3 matrix by a 2 x 2 matrix.
     ::不可能—— 您不能将 2 x 3 矩阵乘以 2 x 2 矩阵 。
  3. $\begin{array}{r}\left[\begin{array}{cc}46& 146\\ 8& 23\end{array}\right]\end{array}$  
  4. $\begin{array}{r}\left[\begin{array}{cc}0& 12\\ 20& 12\\ 4& 24\end{array}\right]\end{array}$  
  5. $\begin{array}{r}\left[\begin{array}{cc}16& 13\\ 4& 10\end{array}\right]\end{array}$  
  6. $\begin{array}{r}\left[\begin{array}{cc}3& \text{-}7\\ \text{-}2& \text{-}6\end{array}\right]\end{array}$  
  7. $\begin{array}{r}\left[\begin{array}{cc}22& 26\\ 6& 16\end{array}\right]\end{array}$  
  8. $\begin{array}{r}\left[\begin{array}{ccc}39& 132& 94\\ 30& 60& 64\end{array}\right]\end{array}$  
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  9. Not possible—you can't multiply a 2 x 3 matrix by a 2 x 2 matrix.
     ::不可能—— 您不能将 2 x 3 矩阵乘以 2 x 2 矩阵 。
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  10. They both equal  $\begin{array}{r}\left[\begin{array}{cc}52& 40\\ 73& 50\end{array}\right]\end{array}$ .
      ::两者相等[52407350]。
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  11. They both equal  $\begin{array}{r}\left[\begin{array}{cc}18& 12\\ 27& 18\end{array}\right]\end{array}$ .
      ::两者相等[18122718]。
  12.   $\begin{array}{r}\left[\begin{array}{ccc}356& 866& 122\\ 528& 653& 285\\ 939& 132& 205\end{array}\right]\end{array}$   
  13. $\begin{array}{r}\left[\begin{array}{ccc}624& 118& 68\\ 684& 312& 378\\ 1,566& 46& 266\end{array}\right]\end{array}$  
  14. $\begin{array}{r}\left[\begin{array}{ccc}132& 288& 84\\ 372& 164& 376\\ 248& 300& 288\end{array}\right]\end{array}$  
  15. $\begin{array}{r}\left[\begin{array}{ccc}21,148& 315,181& 23,377\\ 49,708& 419,179& 29,539\\ 37,679& 672,733& 56,811\end{array}\right]\end{array}$  
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  16. Answers will vary. 
      ::答案将各有不同。
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  17. Friday: $1,019.25; Saturday: $1,295.25; Sunday: $857.75. Total: $3,172.25
      ::星期五:1 019.25美元;星期六:1 295.25美元;星期日:857.75美元。

   

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  Section 10.6: Row Operations and Row Echelon Forms
  ::第10.6节:行操作和行梯表

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  1. Answers will vary.
     ::答案将各有不同。
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  2. Answers will vary.
     ::答案将各有不同。
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  3. Add a multiple of one row to another row; scale a row by multiplying through by a non-zero constant; swap two rows.
     ::将一行的倍数加到另一行;以非零常数乘以一行;交换两行。
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  4. The rows are linearly independent.
     ::各行是线性独立的。
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  5. Answers will vary.
     ::答案将各有不同。
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  6. The rows are linearly independent because reduced row echelon form is the identity matrix.
     ::各行是线性独立的,因为缩小的行梯层表是身份矩阵。
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  7. Answers will vary.
     ::答案将各有不同。
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  8. The rows are linearly independent because reduced row echelon form is the identity matrix.
     ::各行是线性独立的,因为缩小的行梯层表是身份矩阵。
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  9. Answers will vary.
     ::答案将各有不同。
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  10. Reduced row echelon form is  $\begin{array}{r}\left[\begin{array}{cccc}1& 2& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$ . Because there are no zero-only rows, the rows of the original matrix were linearly independent.
      ::缩排梯层表为[1200010010001],由于没有零单行,原始矩阵的行线性独立。
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  11. Answers will vary.
      ::答案将各有不同。
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  12. Reduced row echelon form is $\begin{array}{r}\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\end{array}$ . The rows of matrix D were not linearly independent.
      ::减少的排梯表为[100100]0. 矩阵D各行没有线性独立。
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  13. Answers will vary.
      ::答案将各有不同。
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  14. Reduced row echelon form is $\begin{array}{r}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]\end{array}$ . The rows of matrix E were not linearly independent.
      ::缩排梯层表为[10012000],表格E各行并非线性独立。
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  15. Answers will vary.
      ::答案将各有不同。
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  16. The rows are linearly independent because reduced row echelon form is the identity matrix.
      ::各行是线性独立的,因为缩小的行梯层表是身份矩阵。

   

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  Section 10.7: Augmented Matrices
  ::第10.7节:增加入学人数

  1. (-3, 4)
  2. (1, 6)
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  3. Infinite number of solutions
     ::无限数的解决方案
  4. (3, 2)
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  5. No solution. The lines are parallel and do not intersect.
     ::否。 线条是平行的, 不交叉 。
  6. (1, 4, 6)
  7. (-3, 1, 5)
  8. (2, 5, 3)
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  9. Infinite number of solutions
     ::无限数的解决方案
  10. (3, 2, 8)
  11. (5, -1, 2)
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  12. No solution. If you multiply $\begin{array}{r}{R}_1\end{array}$  by 2 and add it to $\begin{array}{r}{R}_3\end{array}$ , you end up with 0 = 13. Therefore, no solution exists.
      ::否解决方案。如果将R1乘以 2 乘以 2 并添加到 R3, 最终结果为 0 = 13, 因此不存在解决方案 。
  13. (-4, 5, 3)
  14. (1, 6, 8)
  15. (-3, 2, 5)

   

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  Section 10.8: Determinants
  ::第10.8节:决定因素

  1. 2
  2. -27
  3. -4
  4. 1
  5. -22
  6. -9
  7. -89
  8. 294
  9. 8
  10. -186
  11. -56
  12. 88
  13. 124
  14. 176
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  15. Only square matrices have determinants.
      ::只有方格矩阵有决定因素。
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  16. If the determinant is zero, then the rows are not linearly independent.
      ::如果决定因素为零,则行不线性独立。
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  17. No, they are not collinear. Area = 21.5
      ::不,它们不是圆线,面积=21.5。
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  18. No, they are not collinear. Area = 112.5
      ::不,它们不是圆线,面积=112.5
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  19. No, they are not collinear. Area = 35
      ::不,它们不是圆线,面积=35

   

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  Section 10.9: Cramer's Rule
  ::第10.9节:

  1. (-3, 4)
  2. (1, 6)
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  3. There is not one solution because the determinant of the coefficient system is 0. The rows of the coefficient matrix are not linearly independent.
     ::没有一种解决办法,因为系数制度的决定因素是0。系数矩阵的行并非线性独立的。
  4. (3, 2)
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  5. There is not one solution because the determinant of the coefficient system is 0. The rows of the coefficient matrix are not linearly independent.
     ::没有一种解决办法,因为系数制度的决定因素是0。系数矩阵的行并非线性独立的。
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  6. $\begin{array}{r}x=1\end{array}$  
     ::x=1 x=1
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  7. $\begin{array}{r}y=1\end{array}$  
     ::y=1 y=1
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  8. $\begin{array}{r}z=3\end{array}$  
     ::z=3 z=3
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  9. $\begin{array}{r}x=2\end{array}$  
     ::x=2x=2
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  10. $\begin{array}{r}y=2\end{array}$  
      ::y=2 y=2
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  11. $\begin{array}{r}z=2\end{array}$  
      ::z=2 z=2
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  12. There is not one solution because the determinant of the coefficient system is 0. The rows of the coefficient matrix are not linearly independent.
      ::没有一种解决办法,因为系数制度的决定因素是0。系数矩阵的行并非线性独立的。
  13. (-4, 5, 3)
  14. (1, 6, 8)
  15. (-3, 2, 5)
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  16. Look at the other relevant determinants for Cramer's Rule. If they are also zero, then the system has infinite solutions. If they are non-zero, then the system has no solution.
      ::看看Cramer规则的其他相关决定因素。 如果它们也是零, 那么系统就有无限的解决方案。 如果它们不是零, 那么系统就没有解决方案 。
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  17. Paperbacks: $8; Hardcover books: $15
      ::背页: 8美元; 硬封面书本: 15美元
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  18. Corndogs: $2.75; Cotton candies: $1.75
      ::玉米条:2.75美元;棉花糖:1.75美元

   

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  Section 10.10: Inverse Matrices
  ::第10.10节:逆矩阵

  1.  
     $\begin{array}{r}\left[\begin{array}{cc}\frac{3}{2}& \text{-}\frac{5}{2}\\ \text{-}1& 2\end{array}\right]\end{array}$  
  2. 
     $\begin{array}{r}\left[\begin{array}{cc}\text{-}\frac{5}{27}& \frac{2}{9}\\ \frac{2}{27}& \frac{1}{9}\end{array}\right]\end{array}$  
  3. 
     $\begin{array}{r}\left[\begin{array}{cc}0& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{4}\end{array}\right]\end{array}$  
  4. 
     $\begin{array}{r}\left[\begin{array}{cc}1& \text{-}6\\ 0& 1\end{array}\right]\end{array}$  
  5. 
     $\begin{array}{r}\left[\begin{array}{cc}\frac{1}{11}& \frac{5}{22}\\ \frac{2}{22}& \text{-}\frac{3}{11}\end{array}\right]\end{array}$  
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  6. No inverse
     ::无反反
  7. 
     $\begin{array}{r}\left[\begin{array}{ccc}\text{-}\frac{8}{41}& \frac{7}{41}& \frac{5}{41}\\ \frac{19}{41}& \frac{9}{41}& \text{-}\frac{17}{41}\\ \text{-}\frac{6}{41}& \text{-}\frac{5}{41}& \frac{14}{41}\end{array}\right]\end{array}$  
  8. 
     $\begin{array}{r}\left[\begin{array}{ccc}\text{-}\frac{3}{294}& \frac{34}{294}& \frac{5}{294}\\ \frac{18}{294}& \text{-}\frac{8}{294}& \frac{68}{294}\\ \frac{27}{294}& \text{-}\frac{12}{294}& \text{-}\frac{45}{294}\end{array}\right]\end{array}$  
  9. 
     $\begin{array}{r}\left[\begin{array}{ccc}\text{-}1& \text{-}1& \frac{1}{2}\\ \frac{3}{4}& \frac{3}{4}& \text{-}\frac{1}{4}\\ \frac{17}{4}& \frac{21}{4}& \text{-}\frac{7}{4}\end{array}\right]\end{array}$  
  10. 
      $\begin{array}{r}\left[\begin{array}{ccc}\frac{40}{186}& \frac{24}{186}& \text{-}\frac{22}{186}\\ \text{-}\frac{5}{186}& \text{-}\frac{3}{186}& \frac{26}{186}\\ \text{-}\frac{12}{186}& \frac{30}{186}& \text{-}\frac{12}{186}\end{array}\right]\end{array}$  
  11. 
      $\begin{array}{r}\left[\begin{array}{ccc}\frac{14}{56}& \frac{12}{56}& \frac{58}{56}\\ 0& \text{-}\frac{16}{56}& \text{-}\frac{8}{56}\\ \text{-}\frac{7}{56}& \frac{6}{56}& \text{-}\frac{4}{56}\end{array}\right]\end{array}$  
  12. 
      $\begin{array}{r}\left[\begin{array}{ccc}\frac{1}{22}& 0& \text{-}\frac{3}{22}\\ \text{-}\frac{1}{44}& \frac{1}{4}& \frac{3}{44}\\ \frac{9}{22}& \text{-}\frac{1}{2}& \text{-}\frac{5}{22}\end{array}\right]\end{array}$  
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  13. Students should show that the matrix times itself equals the identity matrix.
      ::学生应表明矩阵本身与身份矩阵的比值。
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  14. Students should show that the matrix times itself equals the identity matrix.
      ::学生应表明矩阵本身与身份矩阵的比值。
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  15. A non-square matrix c anno t be multiplied on both sides by the same matrix because the order of the matrices would not work. Therefore, for a non-square matrix there cannot exist just one inverse matrix.
      ::非平方矩阵不能在双方以同一矩阵乘以同一矩阵,因为该矩阵的顺序不会起作用,因此,对于非平方矩阵,不可能只存在一个反向矩阵。

   

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  Section 10.11: Partial Fraction Decomposition
  ::第10.11节:部分分数分数分解

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  1. 
     $\begin{array}{r}\text{-}\frac{\frac{1}{5}}{x-1}+\frac{\frac{16}{5}}{x+4}\end{array}$  
     ::-15x-1+165x+4
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  2. 
     $\begin{array}{r}\text{-}\frac{\frac{7}{9}}{x}-\frac{\frac{1}{3}}{x^2}+\frac{\frac{7}{9}}{x-3}\end{array}$  
     ::-79x-13x2+79x-3
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  3. 
     $\begin{array}{r}\text{-}\frac{\frac{1}{5}}{x}+\frac{\frac{6}{5}}{x-5}\end{array}$  
     ::-15x+65x-5
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  4. 
     $\begin{array}{r}\frac{\text{-}\frac{1}{18}}{x}+\frac{\frac{19}{54}}{x+6}+\frac{\frac{19}{27}}{x-3}\end{array}$  
     ::-118x+1954x+6+1927x-3
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  5. 
     $\begin{array}{r}\text{-}\frac{\frac{1}{2}}{x^2}+\frac{\frac{7}{4}}{x+2}+\frac{\frac{5}{4}}{x}\end{array}$  
     ::-12x2+74x+2+54x
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  6. 
     $\begin{array}{r}\text{-}\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x-1}\end{array}$  
     ::-1x+1x+1+1+1x-1
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  7. 
     $\begin{array}{r}\frac{\frac{9}{4}}{x^2}+\frac{\frac{9}{16}}{x}+\frac{\frac{55}{16}}{x-4}\end{array}$  
     ::94x2+916x+5516x-4
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  8. 
     $\begin{array}{r}\frac{\frac{9}{5}}{x+7}+\frac{\frac{1}{5}}{x-3}\end{array}$  
     ::95x+7+15x-3
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  9. 
     $\begin{array}{r}\frac{4-3x}{x^2+1}-\frac{4}{x^2}+\frac{3}{x}\end{array}$  
     ::4-3x22+1-4x2+3x
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  10. 
      $\begin{array}{r}\frac{\frac{\text{-}11x-7}{13}}{x^2+4}+\frac{\frac{11}{13}}{x-3}\end{array}$  
      ::- 11x-713x2+4+1113x-3
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  11. 
      $\begin{array}{r}\frac{\frac{\text{-}x-13}{5}}{x^2+1}+\frac{\frac{5}{3}}{x^2}+\frac{\frac{14}{45}}{x-3}-\frac{\frac{1}{9}}{x}\end{array}$  
      ::--135x2+1+53x2+1445x-3-19x
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  12. Students should verify that the partial fractions sum to the original function.
      ::学生应核实部分分数与原功能之和。
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  13. Students should verify that the partial fractions sum to the original function.
      ::学生应核实部分分数与原功能之和。
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  14. Students should verify that the partial fractions sum to the original function.
      ::学生应核实部分分数与原功能之和。
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  15. Students should verify that the partial fractions sum to the original function.
      ::学生应核实部分分数与原功能之和。