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Check that they all have the same set of eigenvalues (which are not distinct). (b) Show that A, has r linearly independent eigenvectors for r = 1,2, 3. (In the case of A3, the "equations" don't tell you anything useful-what does that mean in terms of solutions?) [b 1 0 6 1 0 0 b (c) (Extra for fun.) If B show that %| п(п-1) nbr-1 %3D where b is any real number. You could use mathematical induction. If you're not sure about that, at least check the formula works for n = 2,3, 4.

Check that they all have the same set of eigenvalues (which are not distinct). (b) Show that A, has r linearly independent eigenvectors for r = 1,2, 3. (In the case of A3, the "equations" don't tell you anything useful-what does that mean in terms of solutions?) [b 1 0 6 1 0 0 b (c) (Extra for fun.) If B show that %| п(п-1) nbr-1 %3D where b is any real number. You could use mathematical induction. If you're not sure about that, at least check the formula works for n = 2,3, 4.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Consider the 3 matrices 2 1 0 0 2 1 0 0 2 2 1 0 0 2 0 0 0 2 20 0 0 2 0 0 0 2 A1 = A2 A3 : (a) Check that they all have the same set of eigenvalues (which are not distinct). (b) Show that A, has r linearly independent eigenvectors for r = 1,2, 3. (In the case of A3, the "equations" don't tell you anything useful-what does that mean in terms of solutions?) [b 1 0 0 b 1 0 0 b (c) (Extra for fun.) If B = show that bn nbn-1 n(n-1)zn-2" B" = nbn-1 where b is any real number. You could use mathematical induction. If you're not sure about that, at least check the formula works for n = 2, 3, 4.
Transcribed Image Text:1. Consider the 3 matrices 2 1 0 0 2 1 0 0 2 2 1 0 0 2 0 0 0 2 20 0 0 2 0 0 0 2 A1 = A2 A3 : (a) Check that they all have the same set of eigenvalues (which are not distinct). (b) Show that A, has r linearly independent eigenvectors for r = 1,2, 3. (In the case of A3, the "equations" don't tell you anything useful-what does that mean in terms of solutions?) [b 1 0 0 b 1 0 0 b (c) (Extra for fun.) If B = show that bn nbn-1 n(n-1)zn-2" B" = nbn-1 where b is any real number. You could use mathematical induction. If you're not sure about that, at least check the formula works for n = 2, 3, 4.
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