1. 

1. (b) $\left[\begin{array}{c}I\\ A^{-1}\end{array}\right]$;

2. (c) $\left[\begin{array}{cc}{A}^{T}A& A^T\\ A& I\end{array}\right]$;

3. (d) $\begin{array}{c}A{A}^T\end{array}+I$;

4. (e) $\left[\begin{array}{cc}I& A^{-1}\\ A& I\end{array}\right]$

2. Hint: Partition $A^T$ into row vectors and A into column vectors and perform the block multiplication.

3. 

1. (a) $A{\mathbf{b}}_1=\left[\begin{array}{c}3\\ 3\end{array}\right],A{\mathbf{b}}_2=\left[\begin{array}{r}4\\ -1\end{array}\right]$;

2. (b)$\begin{array}{l}[\begin{array}{cc}1& 1\end{array}]B=[\begin{array}{cc}3& 4\end{array}],\\ [\begin{array}{cc}2& -1\end{array}]B=[\begin{array}{cc}3& -1\end{array}];\end{array}$

3. (c) $\mathit{\text{AB}}=\left[\begin{array}{cc}3& \hfill 4\\ 3& \hfill -1\end{array}\right]$

4.

5.

6. 

1. (b) Hint: For a pair of vectors x and y the transpose of the outer product ${\mathbf{\text{xy}}}^T$ is

   $$({\text{xy}}^T{)}^T=({\text{y}}^T{)}^T{\text{x}}^T={\text{yx}}^T$$

8. Hint: AX and B are equal if and only if their corresponding column vectors are equal.

9. 

1. (b) Hint: $A{\text{e}}_j={\text{a}}_j\text{for}j=1,\dots ,n\mathrm{.}$

10. 

1. (b) Hint: Let $X=U\mathrm{\Sigma }\mathrm{.}$ If $A=U\mathrm{\Sigma }{V}^T$ then $A=X{V}^T,$ so A can be expressed as an outer product expansion of $X{V}^T\mathrm{.}$ How are the column vectors of X related to the column vectors of U?

11. Hint: You need to determine a matrix C so that the block multiplication of

will equal the $2n\times 2n$ identity matrix

Once you have found C check that the product of the matrices in the reverse order will also equal $I_{2n}$.

13.
$\begin{array}{cc}{A}^2=\left[\begin{array}{cc}B& O\\ O& B\end{array}\right],& A^4=\left[\begin{array}{cc}{B}^2& O\\ O& B^2\end{array}\right]\end{array}$

14. 

1. (a) $\left[\begin{array}{cc}O& I\\ I& O\end{array}\right]$;

2. (b) $\left[\begin{array}{cc}\hfill I& O\\ \hfill -B& I\end{array}\right]$

15. Hint: The block structure of A resembles the form of an elementary matrix of type III.

16. Hint: The block form of $S^{-1}$ is

17. Hint: Just plug in for B and C and multiply things out.

18. Hint: In order for the block multiplication to work we must have

Fortunately both B and M are nonsingular.

21. Hint: The hint for Exercise20 also applies to this exercise.