Monthly Archives: January 2011

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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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 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

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Linear Algebra and Its Applications, Exercise 1.5.9

Exercise 1.5.9. (a) Assume that A is the product of three matrices as follows: What must be true for A to be nonsingular? (b) Given A above, assume we have a system of linear equations Ax = b with corresponding … Continue reading

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Linear Algebra and Its Applications, Exercise 1.5.8

Exercise 1.5.8. This exercise continues the discussion in p. 33-34 regarding factorization of a 3×3 matrix A and the proof that A = LU. An alternative proof begins with the fact that row 3 of U is produced by taking … Continue reading

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Linear Algebra and Its Applications, Exercise 1.5.7

Exercise 1.5.7. Given the following lower triangular matrices find FGH and HGF. Answer: We haveĀ  where so that We also have where so that Thus . UPDATE: Corrected the calculation of HGF; thanks go to Brian D. for pointing out … Continue reading

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Linear Algebra and Its Applications, Exercise 1.5.6

Exercise 1.5.6. Given the matrix find , , and . Answer: We have We can square this again to obtain and again to obtain In compting we note that all the entries match those of the identity matrix except for … Continue reading

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