1. For the matrix A, see the answers to Exercise 1 of Section 4.2.

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

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

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

4. (d) $B=\left[\begin{array}{cc}\frac{\begin{array}{c}\hfill 1\end{array}}{2}& \hfill 0\\ \hfill 0& \frac{\begin{array}{c}\hfill 1\end{array}}{2}\end{array}\right]$;

5. (e) $B=\left[\begin{array}{cc}\frac{\begin{array}{c}\hfill 1\end{array}}{2}& \frac{\begin{array}{c}\hfill 1\end{array}}{2}\\ \frac{\begin{array}{c}\hfill 1\end{array}}{2}& \frac{\begin{array}{c}\hfill 1\end{array}}{2}\end{array}\right]$

2. 

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

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

3.
$B=A=\left[\begin{array}{ccc}\hfill 2& \hfill -1& \hfill -1\\ \hfill -1& \hfill 2& \hfill -1\\ \hfill -1& \hfill -1& \hfill 2\end{array}\right]$

(Note: in this case the matrices A and U commute; so $B=U^{-1}\mathit{\text{AU}}=U^{-1}\mathit{\text{UA}}=A$.)

4.
$V=\left[\begin{array}{ccc}1& 1& \hfill 0\\ 1& 2& \hfill -2\\ 1& 0& \hfill 1\end{array}\right],B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

5. 

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

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

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

4. (d) $a_{1}x+a_2{2}^n(1+x^2)$

6. 

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

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

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

7. Hint: If A is similar to B then there exists a nonsingular matrix $S_1$ such that $A=S_1^{-1}{\mathit{BS}}_1$ and if B is similar to C then there exists a nonsingular matrix $S_2$ such that $B=S_2^{-1}{\mathit{CS}}_2\mathrm{.}$

9. Hint: Can you use S and $S^{-1}$ to transform A to B?

10. Hint: If A is similar to B then there is a nonsingular matrix S such that

If A is also equal to ST, then how must T be chosen?

11. Hint: If B is similar to A, then $B=S^{-1}\mathit{AS}\mathrm{.}$ Take determinants of both sides of this equation and make use of results from Section 2 of Chapter 2.

14. 

1. (a) Hint: If A and B are similar, then there exists a nonsingular matrix S such that $B={\mathit{SAS}}^{-1}\mathrm{.}$ Look at $S\left(A-\lambda I\right)S^{-1}\mathrm{.}$