Monthly Archives: January 2011

Exercise 1.5.19. For the following two matrices, specify the values of , , and for which elimination requires row exchanges, and the values for which the matrices in question are singular. Answer: For the first matrix a row exchange would … Continue reading →

Exercise 1.5.18. We have the following systems of linear equations: and and Which of the systems are singular, and which nonsingular? Which have no solutions? One solution? An infinite number of solutions? Answer: For the first system we can add … Continue reading →

Exercise 1.5.17. If LPU order is used then rows are exchanged only at the end of elimination: Specify what L is in the above case. Answer: Since no row exchanges are done during elimination proper, the multipliers in L stay … Continue reading →

Exercise 1.5.16. Find a 4 by 4 matrix (preferably a permutation matrix) that is nonsingular and for which elimination requires three row exchanges. Answer: The following permutation matrix meets the requirement: For this matrix elimination requires the following row exchanges: … Continue reading →

Exercise 1.5.15. Given the matrices and find their factors L, D, and U and associated permutation matrix P such that PA = LDU. Confirm that the factors are correct. Answer: In the first step in elimination for the first matrix … Continue reading →

Exercise 1.5.14. Find all possible 3 by 3 permutation matrices, along with their inverses. Answer: The identity matrix I is the first possible permutation matrix, corresponding to not doing a row exchange at all; it is its own inverse: The … Continue reading →

Exercise 1.5.13. Given the systems of equations and solve both by elimination. Do row exchanges where necessary, and specify any permutation matrices required. Answer: The first system of equations can be expressed as follows: In the first step of elimination … Continue reading →

Exercise 1.5.11. We have a system LUx = b with values for L, U, and b as follows: Solve for x without multiplying L and U to find A. Answer: We can take advantage of the equations Lc = b … Continue reading →

Exercise 1.5.10. (a) Both Lc = b and Ux = c take approximately multiplication-substraction steps to solve. Explain why. (b) Assume A is a 60 by 60 coefficient matrix. How many steps are required to use elimination to solve ten … Continue reading →