1. 

1. (a) $\left[\begin{array}{ccc}\hfill 6& 2& 8\\ \hfill -4& 0& 2\\ \hfill 2& 4& 4\end{array}\right]$;

2. (b) $\left[\begin{array}{ccc}\hfill 4& \hfill 1& 6\\ \hfill -5& \hfill 1& 2\\ \hfill 3& \hfill -2& 3\end{array}\right]$;

3. (c) $\left[\begin{array}{ccc}\hfill 3& \hfill 2& \hfill 2\\ \hfill 5& \hfill -3& \hfill -1\\ \hfill -4& \hfill 16& \hfill 1\end{array}\right]$;

4. (d) $\left[\begin{array}{ccc}3& \hfill 5& \hfill -4\\ 2& \hfill -3& \hfill 16\\ 2& \hfill -1& \hfill 1\end{array}\right]$;

5. (f) $\left[\begin{array}{ccc}\hfill 5& \hfill 5& \hfill 8\\ \hfill -10& \hfill -1& \hfill -9\\ \hfill 15& \hfill 4& \hfill 6\end{array}\right]$;

6. (h) $\left[\begin{array}{ccc}5& \hfill -10& \hfill 15\\ 5& \hfill -1& \hfill 4\\ 8& \hfill -9& \hfill 6\end{array}\right]$

2. 

1. (a) $\left[\begin{array}{cc}15& 19\\ 4& 0\end{array}\right]$;

2. (c) $\left[\begin{array}{cc}\hfill 19& 21\\ \hfill 17& 21\\ \hfill 8& 10\end{array}\right]$;

3. (f) $\left[\begin{array}{cccc}\hfill 6& \hfill 4& \hfill 8& \hfill 10\\ \hfill -3& \hfill -2& \hfill -4& \hfill -5\\ \hfill 9& \hfill 6& \hfill 12& \hfill 15\end{array}\right]$

(b) and (e) are not possible.

3. 

1. (a) $3\times 3$;

2. (b) $1\times 2$

4. 

1. (a) $\left[\begin{array}{cc}3& \hfill 2\\ 2& \hfill -3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}1\\ 5\end{array}\right]$;

2. (b) $\left[\begin{array}{ccc}1& \hfill 1& \hfill 0\\ 2& \hfill 1& \hfill -1\\ 3& \hfill -2& \hfill 2\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{x}_1\\ x_2\end{array}\\ x_3\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}5\\ 6\end{array}\\ 7\end{array}\right]$;

3. (c) $\left[\begin{array}{ccc}2& \hfill 1& \hfill 1\\ 1& \hfill -1& \hfill 2\\ 3& \hfill -2& \hfill -1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{x}_1\\ x_2\end{array}\\ x_3\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}4\\ 2\end{array}\\ 0\end{array}\right]$

9. 

1. (a) $\mathbf{b}=2{\mathbf{a}}_1+{\mathbf{a}}_2$

10. 

1. (a) inconsistent;

2. (b) consistent;

3. (c) Hint: If c is any linear combination of the column vectors of A then $c_1=c_2=c_3$.

11. Hint: $\mathbf{b}=1{\mathbf{a}}_1+1{\mathbf{a}}_2+0{\mathbf{a}}_3=0{\mathbf{a}}_1+1{\mathbf{a}}_2+1{\mathbf{a}}_3$.

13. 

1. (b) $\mathbf{b}={(8,-7,-1,7)}^T$

14. $\mathbf{w}={(\frac{1}{2},\frac{1}{3},\frac{1}{6})}^T,\mathbf{r}=(\frac{43}{120},\frac{45}{120},\frac{32}{120}{)}^T$

18. $b=a_{22}-\frac{a_{12}{a}_{21}}{a_{11}}$