Linear Algebra with Applications, 10th Edition

2.3

1. 1. 

   1. (a) $\begin{array}{c}\begin{array}{c}\text{det}(A)=-7,\text{adj }A=\left[\begin{array}{cc}\hfill -1& \hfill -2\\ \hfill -3& \hfill 1\end{array}\right],\end{array}\\ A^{-1}=\left[\begin{array}{cc}\frac{1}{7}& \frac{2}{7}\\ \frac{3}{7}& -\frac{1}{7}\end{array}\right];\hfill \end{array}$

   2. (c) $\begin{array}{c}\begin{array}{c}\text{det}(A)=3,\text{adj }A=\left[\begin{array}{ccc}\hfill -3& \hfill 5& \hfill 2\\ \hfill 0& \hfill 1& \hfill 1\\ \hfill 6& \hfill -8& \hfill -5\end{array}\right],\end{array}\\ A^{-1}=\frac{1}{3}\text{adj }A\hfill \end{array}$

2. 2. 

   1. (a) $(\frac{5}{7},\frac{8}{7})$;

   2. (b) $(\frac{11}{5},-\frac{4}{5})$;

   3. (c) $(4,-2,2)$;

   4. (d) $(2,-1,2)$;

   5. (e) $(-\frac{2}{3},\frac{2}{3},\frac{1}{3},0)$

3. 3. $-\frac{3}{4}$

4. 4. $(\frac{1}{2},-\frac{3}{4},1{)}^T$

5. 5. 

   1. (a) $\text{det}(A)=0$, so A is singular.

   2. (b) $\begin{array}{c}\text{adj }A=\left[\begin{array}{ccc}\hfill -1& \hfill 2& \hfill -1\\ \hfill 2& \hfill -4& \hfill 2\\ \hfill -1& \hfill 2& \hfill -1\end{array}\right]\text{ and}\\ A\text{adj }A=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& \hfill 0\\ 0& 0& 0\end{array}\right]\hfill \end{array}$

6. 7. Hint: The solution of the linear system $I\mathbf{\text{x = b}}\text{ is }\mathbf{\text{x=b.}}$

7. 9. 

   1. (a) $\text{det}(\text{adj}(A))=8$ and $\text{det}(A)=2$;

   2. (b) $A=\left[\begin{array}{cccc}1& \hfill 0& \hfill 0& \hfill 0\\ 0& \hfill 4& \hfill -1& \hfill 1\\ 0& \hfill -6& \hfill 2& \hfill -2\\ 0& \hfill 1& \hfill 0& \hfill 1\end{array}\right]$

8. 12. Hint: If $\text{det}\left(A\right)=1,$ show that

   $$\text{adj}A=A^{-1}$$

   and then apply the result from Exercise10 .

9. 13. Hint: The $(j,i)$ entry of $Q^T$ is $q_{ij}\mathrm{.}$ Determine an expression for the $(j,i)$ entry of $Q^{-1}$ involving a cofactor of Q.

10. 14. Do Your Homework.

11. 15. 

    1. (d) Hint: Expand the determinant into cofactors along the third row.