Linear Algebra with Applications, 10th Edition

Section 7.3 Exercises

1. Let

   $$\begin{array}{cc}\begin{array}{cc}A=\left[\begin{array}{ccc}0& 3& 1\\ 1& 2& \hfill -2\\ 2& 5& \hfill 4\end{array}\right]& \text{and}\end{array}& \mathbf{b}=\left[\begin{array}{r}\begin{array}{c}1\\ 7\end{array}\\ -1\end{array}\right]\end{array}$$

   1. Reorder the rows of (A|b) in the order (2, 3, 1) and then solve the reordered system.

   2. Factor A into a product $P^T\mathit{\text{LU}}$, where P is the permutation matrix corresponding to the reordering in part (a).

2. Let A be the matrix in Exercise 1. Use the factorization $P^T\mathit{\text{LU}}$ to solve $A\mathbf{x}=\mathbf{c}$ for each of the following choices of c:

   1. ${(8,1,20)}^T$

   2. ${(-9,-2,-7)}^T$

   3. ${(4,1,11)}^T$

3. Let

   $$\begin{array}{cc}\begin{array}{cc}A=\left[\begin{array}{ccc}\hfill 1& \hfill 8& \hfill 6\\ \hfill -1& \hfill -4& \hfill 5\\ \hfill 2& \hfill 4& \hfill -6\end{array}\right]& \text{and}\end{array}& \mathbf{b}=\left[\begin{array}{c}\begin{array}{c}8\\ 1\end{array}\\ 4\end{array}\right]\end{array}$$

   Solve the system $A\mathbf{x}=\mathbf{b}$ using partial pivoting. If P is the permutation matrix corresponding to the pivoting strategy, factor PA into a product LU.

4. Let

   $$\begin{array}{cc}\begin{array}{cc}A=\left[\begin{array}{cc}3& 2\\ 2& 4\end{array}\right]& \text{and}\end{array}& \mathbf{b}=\left[\begin{array}{r}5\\ -2\end{array}\right]\end{array}$$

   Solve the system $A\mathbf{x}=\mathbf{b}$ using complete pivoting. Let P be the permutation matrix determined by the pivot rows and Q the permutation matrix determined ...