Linear Algebra with Applications, 10th Edition

1.6

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

3. 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. 4.

5. 5.

6. 6. 

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

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

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

8. 9. 

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

9. 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 ...