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Monthly Archives: January 2011
Linear Algebra and Its Applications, Exercise 1.5.19
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
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Linear Algebra and Its Applications, Exercise 1.5.18
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
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Linear Algebra and Its Applications, Exercise 1.5.17
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
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Linear Algebra and Its Applications, Exercise 1.5.16
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
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Linear Algebra and Its Applications, Exercise 1.5.15
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
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Linear Algebra and Its Applications, Exercise 1.5.14
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
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Linear Algebra and Its Applications, Exercise 1.5.13
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
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Linear Algebra and Its Applications, Exercise 1.5.12
Exercise 1.5.12. Could be factored into the product where is upper triangular and is lower triangular, instead of being factored into the product ? If so, how could this other factorization be carried out? Would and be the same in … Continue reading
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Tagged factorization, lower triangular matrix, upper triangular matrix
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Linear Algebra and Its Applications, Exercise 1.5.11
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
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Linear Algebra and Its Applications, Exercise 1.5.10
Exercise 1.5.10. (a) Both Lc = b and Ux = c take approximately multiplicationsubstraction 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
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