Due: 03/15/2005

- [1.] 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}as well as M^{-1. } -
[2.] 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. - [3.] Problem 19 (page 44)
- [4.] Problem 17 (page 54)
- [5.] Problem 30 (page 56)
- [6.] Problem 13 (page 67)
- [7.] Problem 15 (page 67)
- [8.] 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.
- [9.] Problem 10 (page 79)
- [10.] Problem 29 (page 81)
- [11.] Problem 32 (page 82)
- [12.] Problem 13 (page 93)
- [13.] Problem 11 (page 105)
- [14.] Problem 12 (page 107)
- [15.] Problem 31 (page 108)
- [16.] Problem 41 (page 109)