Linear Algebra with Applications, 10th Edition

1.1

1. 1. 

   1. (a) $(11,3)$

   2. (b) $(4,1,3)$;

   3. (c) $(-2,0,3,1)$;

   4. (d) $(-2,3,0,3,1)$

2. 2. 

   1. (a) $\left[\begin{array}{cc}1& \hfill -3\\ 0& \hfill 2\end{array}\right]$;

   2. (b) $\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 1\\ 0& 0& 3\end{array}\right]$;

   3. (c) $\left[\begin{array}{cccc}1& 2& \hfill 2& \hfill 1\\ 0& 3& \hfill 1& \hfill -2\\ 0& 0& \hfill -1& \hfill 2\\ 0& 0& \hfill 0& \hfill 4\end{array}\right]$

3. 3. 

   1. (a) One solution. The two lines intersect at the point $(3,1)$.

   2. (b) No solution. The lines are parallel.

   3. (c) Infinitely many solutions. Both equations represent the same line.

   4. (d) No solution. Each pair of lines intersect in a point; however, there is no point that is on all three lines.

4. 4. 

   1. (a) $\left[\begin{array}{cc}1& \hfill 1\\ 1& \hfill -1\end{array}\right|\begin{array}{c}4\\ 2\end{array}]$;

   2. (c) $\left[\begin{array}{cc}\hfill 2& \hfill -1\\ \hfill -4& \hfill 2\end{array}\right|\begin{array}{r}3\\ -6\end{array}]$;

   3. (d) $\left[\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 1& \hfill -1\\ \hfill -1& \hfill 3\end{array}\right|\begin{array}{c}\begin{array}{c}1\\ 1\end{array}\\ 3\end{array}]$

5. 6. 

   1. (a) $(1,-2)$;

   2. (b) $(3,2)$;

   3. (c) $(\frac{1}{2},\frac{2}{3})$;

   4. (d) $(1,1,2)$;

   5. (e) $(-3,1,2)$;

   6. (f) $(-1,1,1)$

   7. (g) $(1,1,-1)$;

   8. (h) $(4,-3,1,2)$;

6. 7. 

   1. (a) $(2,-1)$;

   2. (b) $(-2,3)$

7. 8. 

   1. (a) $(-1,2,1)$;

   2. (b) $(3,1,-2)$

8. 9. Hints: 

   1. (a) and (b)

      Subtract the first equation from the second. Now replace the second ...