# CAS CS 232 Geometric Algorithms A1

## 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 E2,1, E3,1, that put A into triangular form U with E3,2 E3,1E2,1.

Compute M = E3,2 E3,1E2,1.

• [3.] 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.
• [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)