Elementary Linear Algebra, 11th Edition

ANSWERS TO EXERCISES

Exercise Set 1.1

1. 1. (a), (c), and (f) are linear equations; (b), (d), and (e) are not linear equations
2. 3.(a) $\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2=b_1\\ a_{21}{x}_1+a_{22}{x}_2=b_2\end{array}$
   1. (b)$\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3=b_1\\ a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3=b_2\\ a_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3=b_3\end{array}$
   2. (c) $\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3+a_{14}{x}_4=b_1\\ a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3+a_{24}{x}_4=b_2\end{array}$
3. 5.(a) $\begin{array}{r}\hfill 2x_1=0\\ \hfill 3x_1-4x_2=0\\ \hfill x_2=1\end{array}$
   1. (b) $\begin{array}{l}3x_1-2x_3=5\\ 7x_1+x_2+4x_3=-3\\ -2x_2+x_3=7\end{array}$
4. 7. (a) $\left[\begin{array}{rr}\hfill -2& \hfill 6\\ \hfill 3& \hfill 8\\ \hfill 9& \hfill -3\end{array}\right]$
   1. (b) $\left[\begin{array}{rrrr}\hfill 6& \hfill -1& \hfill 3& \hfill 4\\ \hfill 0& \hfill 5& \hfill -1& \hfill 1\end{array}\right]$
   2. (c) $\left[\begin{array}{rrrrrr}\hfill 0& \hfill 2& \hfill 0& \hfill -3& \hfill 1& \hfill 0\\ \hfill -3& \hfill -1& \hfill 1& \hfill 0& \hfill 0& \hfill -1\\ \hfill 6& \hfill 2& \hfill -1& \hfill 2& \hfill -3& \hfill 6\end{array}\right]$
5. 9. (a), (d), and (e) are solutions; (b) and (c) are not solutions
6. 11. (a) No points of intersection
   1. (b) Infinitely many points of intersection: $x=\frac{1}{2}+2t,y=t$
   2. (c) One point of intersection: $\left(-8,-4\right)$
7. 13.(a) $x=\frac{3}{7}+\frac{5}{7}t,y=t$
   1. (b) $x_1=\frac{7}{3}+\frac{5}{3}r-\frac{4}{3}s,x_2=r,x_3=s$
   2. (c) $x_1=-\frac{1}{8}+\frac{1}{4}r-\frac{5}{8}s+\frac{3}{4}t,x_2=r,x_3=s,x_4=t$
   3. (d) $v=\frac{8}{3}{t}_1-\frac{2}{3}{t}_2+\frac{1}{3}{t}_3-\frac{4}{3}{t}_4,w=t_1,x=t_2,y=t_3,z=t_4$
8. 15.(a) $x=\frac{1}{2}+\frac{3}{2}t,y=t$
   1. (b) $x_1=-4-3r+s,x_2=r,x_3=s$
9. 17. (a) Add 2 times the second row to the first row.
   1. (b) Add the third row to the first row, or interchange the first row and the third row.
10. 19. (a) All values of $k\ne 2$
    1. (b) All values of $k$
11. 25. $\begin{array}{c}2x+3y+z=7\\ 2x+y+3z=9\\ 4x+2y+5z=16\end{array}$
12. 27. $\begin{array}{rr}\hfill x+y+z=& \hfill 12\\ \hfill 2x+y+2z=& \hfill 5\\ \hfill -x+z=& \hfill 1\end{array}$

True/False 1.1

1. (a) True
2. (b) False
3. (c) True
4. (d) True
5. (e) False
6. (f) False
7. (g) True
8. (h) False

Exercise Set 1.2

1. 1. (a) Both
   1. (b) Both
   2. (c) Both
   3. (d) Both
   4. (e) Both
   5. (f) Both
   6. (g) Row echelon form
2. 3. (a) $x=-37,y=-8,z=5$
   1. (b) $w=-10+13t,x=-5+13t,y=2-t,z=t$
   2. (c) $x_1=-11-7s+2t,x_2=s,x_3=-4-3t,x_4=9-3t,x_5=t$
   3. (d) No solution
3. 5. $x_1=3,x_2$