Monthly Archives: May 2010

Exercise 1.3.11. Given the systems of equations    and    solve both systems using Gaussian elimination. Answer: We start with the first system of equations The first elimination step produces The second elimination step produces We then back-substitute, starting with … Continue reading →

Exercise 1.3.10 (very optional). Find a method for computing the quantities ac – bd and bc + ad with three multiplications instead of four. Assuming that addition were a sufficiently faster operation than multiplication, this would provide a faster way … Continue reading →

Exercise 1.3.9. State whether the following statements are true or false. (Note that without loss of generality we can assume that no row exchanges occur during the process of elimination.) (a) Given a system in u, v, etc., where the … Continue reading →

Exercise 1.3.8. Given a system of equations of order n = 600, how long would it take to solve in terms of the number of multiplication-subtractions? In seconds, on a PC capable of 8,000 operations per second? On a VAX … Continue reading →

Exercise 1.3.7. (a) Given a system of equations A, with the first two rows the same, at what point in elimination will it become clear that A is singular? Show a 3×3 example. (b) Repeat (a), but instead assume that … Continue reading →

Exercise 1.3.6. Given the following system of equations: for what values of would Gaussian elimination break down, either in a fixable way (i.e., via row exchange) or in a non-fixable way? Answer: One way for elimination to break down temporarily … Continue reading →

Exercise 1.3.5. Given the following system of equations: find a solution to the system. (The exercise also calls for doing a sketch of the lines corresponding to the equations, as well as the line corresponding to the second equation after … Continue reading →

Exercise 1.3.4. Given the following system of equations: find a solution to the system, exchanging rows when necessary due to a zero pivot. Also, specify a coefficient for v in the third equation that would prevent elimination from being successful. … Continue reading →

Exercise 1.3.3. Given the following system of equations: find a solution to the system, and give the pivots. You can use a matrix to represent the system (including the right-hand side). Answer: We can represent the system as the following … Continue reading →

Exercise 1.3.2. Perform Gaussian elimination on the following system of equations: and find the resulting triangular system and the solution. Answer: The first pivot is 1. We subtract the first equation from the second and the third: The second pivot … Continue reading →