Linear Algebra with Applications, 10th Edition

6.7

1. 1. 

   1. (a) $\text{det}(A_1)=2,\text{det}(A_2)=3$, positive definite;

   2. (b) $\text{det}(A_1)=3,\text{det}(A_2)=-10$, not positive definite;

   3. (c) $\text{det}(A_1)=6,\text{det}(A_2)=14,\text{det}(A_3)=-38$, not positive definite;

   4. (d) $\text{det}(A_1)=4,\text{det}(A_2)=8,\text{det}(A_3)=13,$, positive definite

2. 2. $a_{11}=3,a_{22}^{(1)}=2,a_{33}^{(2)}=\frac{4}{3}$

3. 4. 

   1. (a) $\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]\left[\begin{array}{cc}4& 0\\ 0& 9\end{array}\right]\left[\begin{array}{cc}1& \frac{1}{2}\\ 0& 1\end{array}\right]$;

   2. (b) $\left[\begin{array}{cc}\hfill 1& 0\\ \hfill -\frac{1}{3}& 1\end{array}\right]\left[\begin{array}{cc}9& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& \hfill -\frac{1}{3}\\ 0& \hfill 1\end{array}\right]$;

   3. (c) $\left[\begin{array}{ccc}1& \hfill 0& 0\\ \frac{1}{2}& \hfill 1& 0\\ \frac{1}{4}& \hfill -1& 1\end{array}\right]\left[\begin{array}{ccc}16& 0& 0\\ 0& 2& 0\\ 0& 0& 4\end{array}\right]\left[\begin{array}{ccc}1& \frac{1}{2}& \hfill \frac{1}{4}\\ 0& 1& \hfill -1\\ 0& 0& \hfill 1\end{array}\right]$;

   4. (d) $\left[\begin{array}{ccc}1& \hfill 0& 0\\ \frac{1}{3}& \hfill 1& 0\\ -\frac{2}{3}& \hfill 1& 1\end{array}\right]\left[\begin{array}{ccc}9& 0& 0\\ 0& 3& 0\\ 0& 0& 2\end{array}\right]\left[\begin{array}{ccc}1& \frac{1}{3}& \hfill -\frac{2}{3}\\ 0& 1& \hfill 1\\ 0& 0& \hfill 1\end{array}\right]$

4. 5. 

   1. (a) $\left[\begin{array}{cc}2& 0\\ 1& 3\end{array}\right]\left[\begin{array}{cc}2& 1\\ 0& 3\end{array}\right]$;

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

   3. (c) $\left[\begin{array}{ccc}4& \hfill 0& \hfill 0\\ 2& \hfill \sqrt{2}& \hfill 0\\ 1& \hfill -\sqrt{2}& \hfill 2\end{array}\right]\left[\begin{array}{ccc}4& \hfill 2& \hfill 1\\ 0& \hfill \sqrt{2}& \hfill -\sqrt{2}\\ 0& \hfill 0& \hfill 2\end{array}\right]$;

   4. (d) $\left[\begin{array}{ccc}\hfill 3& \hfill 0& \hfill 0\\ \hfill 1& \hfill \sqrt{3}& \hfill 0\\ \hfill -2& \hfill \sqrt{3}& \sqrt{\begin{array}{c}\hfill 2\end{array}}\end{array}\right]\left[\begin{array}{ccc}3& 1& \hfill -2\\ 0& \hfill \sqrt{3}& \hfill \sqrt{3}\\ 0& \hfill 0& \sqrt{\begin{array}{c}\hfill 2\end{array}}\end{array}\right]$

5. 6. Hint: You must show that the three conditions in the definition of an inner product are all satisfied.

6. 7. 

   1. (a) Hint: We compute the inverse of U by using row operations to accomplish the transformation

      $$\left[\begin{array}{cc}U& I\end{array}\right]\to [\begin{array}{c}I\end{array}$$