# CAS CS 232 Geometric Algorithms A1

## Due: 03/15/2005

• [1.] Problem 12 (page 43).
In addition, let Ax = b denote the above system, write the three elimination matrices E2,1, E3,1, that put A into triangular form U with E3,2 E3,1E2,1.

Compute M = E3,2 E3,1E2,1 as well as M -1.

• [2.] Consider the 3 elimination matrices of the previous exercise. Which of the products
E2,1 E3,1,
E2,1 E3,2
E3,1 E3,2
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)