Math 206 Graded Problems/Proofs #4 Due Thursday February 14

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Follow the usual instructions when you write the final draft of your solutions.

• Let A, D, and P be n x n matrices satisfying P^-1AP = D.
  Solve this equation for A. Must it be true that A = D? Show that P^-1A²P = D².
  
  
  
  
  
  
  

• Let A be the 3 x 3 matrix

  0	0	-2
  1	2	1
  1	0	3

  
  Let P be the invertible 3 x 3 matrix

  -1	0	-2
  0	1	1
  1	0	1

  
  Find P^-1 A P and thereby verify that P^-1A P is a diagonal matrix (one whose nonzero entries, if any, all appear on the main diagonal). Use this, together with ideas based on Problem 1, to find A² and A¹⁰ without working any harder than you have to. You should do this entire problem without using computer-algebra software or a calculator (except for number-arithmetic).

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: February 7, 2002
• Expires: June 30, 2002