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

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

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

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

7. 

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

   $$\left[\begin{array}{cc}U& I\end{array}\right]\to \left[\begin{array}{cc}I& U^{-1}\end{array}\right]$$

   The row operations are chosen to zero out the entries above the diagonal of the left matrix in the partition. What effect do these operations have on the right matrix in the partition?

2. (b) The $(i,j)$ entry of $U_1{U}_2$ is computed using the ith row vector of $U_1$ and the jth column vector of $U_2$. How do these two vectors pair up when $j<i?$, when $j=i?$

8. Hint: Use the hint in the book and show that $U_2{U}_1^{-1}=I\mathrm{.}$

9. Hint: Suppose A has two Cholesky factorizations

where L and K are $n\times n$ lower triangular matrices with positive diagonal entries. Factor these matrices into products

where $L_1,L_2$ are unit lower triangular matrices and $D_1,D_2$ are diagonal matrices and then form two LDU factorizations of the matrix A. Make use of the uniqueness of the LDU factorization to show that L and K must be equal.

10. Hint: Show first that if A is an $m\times n$ matrix of rank n and x is a nonzero vector in ${ℝ}^n,$ then $\mathbf{\text{y =}}A\mathbf{\text{x}}$ is a nonzero vector in ${ℝ}^m\mathrm{.}$

11. Hint: $A^{T}A=\left(QR\right)(QR{)}^T=R^T{Q}^{T}QR\mathrm{.}$ Since the matrix Q has orthonormal column vectors, the $(i,j)$ entry of $Q^{T}Q$ will be ${\mathbf{\text{q}}}_i^T{\mathbf{\text{q}}}_j={\delta }_{ij\mathrm{.}}$

13. Hint: Use the definition of the matrix exponential to show $(e^A{)}^T=e^A\mathrm{.}$ How are the eigenvalues of A and $e^A$ related?

14. Hint: If B is symmetric then $B^2=B^{T}B\mathrm{.}$

15. 

1. (b) Hint: To show $B^2$ is not positive definite you must find a nonzero vector x such that ${\mathbf{\text{x}}}^T{B}^2\mathbf{\text{x}}\le 0.$

16. 

1. (b) Hint: The proof is similar to the proof that the leading principal sub-matrices of a positive definite matrix are all positive definite.