Due: 02/12/2004

- [1.] Problem 7 (page 42).
- [2.] Problem 12 (page 43).

In addition, let Ax = b denote the above system, write the three elimination matrices E_{2,1}, E_{3,1}, that put A into triangular form U with E_{3,2}E_{3,1}E_{2,1}.Compute M = E

_{3,2}E_{3,1}E_{2,1}. - [3.] Consider the 3 elimination matrices of the previous exercise.
Which of the products

E_{2,1}E_{3,1},

E_{2,1}E_{3,2}

E_{3,1}E_{3,2}

commute (the product AC commutes if AB = BA). Explain, you actually do not need to compute the products to answer this question, you can simply reason about what elimination matrices do. - [4.] Problem 20 (page 44)
- [5.] Problem 21 (page 44)
- [6.] Problem 3 (page 53) with A(3,1) changed from -2 to -4
- [7.] Problem 14 (page 54)
- [8.] Problem 17 (page 54)
- [9.] Problem 26 (page 55)
- [10.] Problem 6 (page 66)
- [11.] Problem 15 (page 67)
- [12.] Let A be a 100 by 100 square matrix. Assume that a system of equations Ax = b has a unique solution. Is it possible that the system of equations Ax = c for some other right-hand-side c has either no solutions or infinitely many? Please explain your reasoning clearly (Hint: think of what happens when you use elimination on the augmented matrix.
- [13.] Problem 10 (page 79)
- [14.] Problem 23 (page 80)
- [15.] Problem 29 (page 81)
- [16.] Problem 32 (page 82)