Linear Algebra with Applications, 10th Edition

Section 6.7 Exercises

1. For each of the following matrices, compute the determinants of all the leading principal submatrices and use them to determine whether the matrix is positive definite:

   1. $\left[\begin{array}{cc}\hfill 2& \hfill -1\\ \hfill -1& \hfill 2\end{array}\right]$

   2. $\left[\begin{array}{cc}\hfill 3& \hfill 4\\ \hfill 4& \hfill 2\end{array}\right]$

   3. $\left[\begin{array}{ccc}\hfill 6& 4& \hfill -2\\ \hfill 4& 5& \hfill 3\\ \hfill -2& 3& \hfill 6\end{array}\right]$

   4. $\left[\begin{array}{ccc}\hfill 4& \hfill 2& \hfill 1\\ \hfill 2& \hfill 3& \hfill -2\\ \hfill 1& \hfill -2& \hfill 5\end{array}\right]$

2. Let A be a $3\times 3$ symmetric positive definite matrix and suppose that $\text{det}\left(A_1\right)=3,\text{det}\left(A_2\right)=6$, and $\text{det}\left(A_3\right)=8$. What would the pivot elements be in the reduction of A to triangular form, assuming that only row operation III is used in the reduction process?

3. Let

   $$A=\left[\begin{array}{cccc}\hfill 2& \hfill -1& \hfill 0& \hfill 0\\ \hfill -1& \hfill 2& \hfill -1& \hfill 0\\ \hfill 0& \hfill -1& \hfill 2& \hfill -1\\ \hfill 0& \hfill 0& \hfill -1& \hfill 2\end{array}\right]$$

   1. Compute the LU factorization of A.

   2. Explain why A must be positive definite.

4. For each of the following, factor the given matrix into a product ${\mathit{\text{LDL}}}^T$, where L is lower triangular with 1’s on the diagonal and D is a diagonal matrix: ...