1.

1. (a) T

2. (b) T

3. (c) F

4. (d) T

5. (e) T

6. (f) F

7. (g) F

8. (h) T

2. $J=\left(\begin{array}{ccc}{A}_1& O& O\\ O& A_2& O\\ O& O& A_3\end{array}\right)$ where $A_1=\left(\begin{array}{cccccc}2& 1& 0& 0& 0& 0\\ 0& 2& 1& 0& 0& 0\\ 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 2& 1& 0\\ 0& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 2\end{array}\right),$

$A_2=\left(\begin{array}{rrrr}4& 1& 0& 0\\ 0& 4& 1& 0\\ 0& 0& 4& 0\\ 0& 0& 0& 4\end{array}\right)$ and $A_3=\left(\begin{array}{rr}-3& 0\\ 0& -3\end{array}\right)$

3.

1. (a) $-{(t-2)}^5{(t-3)}^2$

2. (b)

   

   1-1 Full Alternative Text

3. (c) ${\lambda }_3=3$

4. (d) $p_1=3\text{ and }{p}_2=1$

5. (e)

   1. (i) $\text{rank}({\text{U}}_1)=3\text{ and rank}({\text{U}}_2)=0$

   2. (ii) $\text{rank}\left({{\displaystyle \text{U}}}_1^2\right)=1\text{ and rank}\left({{\displaystyle \text{U}}}_2^2\right)=0$

   3. (iii) $\text{nullity}({\text{U}}_1)=2\text{ and nullity}({\text{U}}_2)=2$

   4. (iv) $\text{nullity}\left({{\displaystyle \text{U}}}_1^2\right)=4\text{ and nullity}\left({{\displaystyle \text{U}}}_2^2\right)=2$

4.

1. (a) $J=\left(\begin{array}{rrr}1& 0& 0\\ 0& 2& 1\\ 0& 0& 2\end{array}\right)\text{and}Q=\left(\begin{array}{rrr}1& 1& 1\\ 2& 1& 2\\ 1& -1& 0\end{array}\right)$

2. (d) $J=\left(\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right)\text{and}Q=\left(\begin{array}{rrrr}1& 0& 1& -1\\ 1& -1& 0& 1\\ 1& -2& 0& 1\\ 1& 0& 1& 0\end{array}\right)$

5.

1. (a) $J=\left(\begin{array}{rrrr}1& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 2\end{array}\right)\text{and}\beta =\{2e^t,2t{e}^t,t^2{e}^t,e^{2t}\}$

2. (c) $J=\left(\begin{array}{rrrr}2& 1& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 2\end{array}\right)\text{and}\beta =\{6x,x^3,2,x^2\}$

3. (d) $\begin{array}{rcl}J& =& \left(\begin{array}{rrrr}2& 1& 0& 0\\ 0& 2& 1& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 4\end{array}\right)\text{and}\\ \beta & =& \left\{\left(\begin{array}{rr}1& 0\\ 0& 0\end{array}\right),\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right),\left(\begin{array}{rr}0& -1\\ 0& 2\end{array}\right),\left(\begin{array}{rr}1& -2\\ 2& 0\end{array}\right)\right\}\end{array}$

24.

1. (a) $\left(\begin{array}{r}x\\ y\\ z\end{array}\right)=e^{2t}\left[(c_1+c_{2}t)\left(\begin{array}{r}1\\ 0\\ 0\end{array}\right)+c_2\left(\begin{array}{r}0\\ 1\\ 0\end{array}\right)\right]+c_3{e}^{3t}\left(\begin{array}{r}1\\ 1\\ -1\end{array}\right)$

2. (b) $\left(\begin{array}{r}x\\ y\\ z\end{array}\right)=e^{2t}\left[(c_1+c_{2}t+c_3{t}^2)\left(\begin{array}{r}1\\ 0\\ 0\end{array}\right)+(c_2+2c_{3}t)\left(\begin{array}{r}0\\ 1\\ 0\end{array}\right)+2c_3\left(\begin{array}{r}0\\ 0\\ 1\end{array}\right)\right]$