anonymous
  • anonymous
Hey everyone..P, A, and B are nXn matrices and have the equation B=P^-1AP (so they are similar) Show that B^2=P^-1A^2P and find B^k and A^k
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
im new plz tell me what does ^ means
anonymous
  • anonymous
It means "to the exponent of" so whatever comes before that symbol is raised to the exponent of whatever comes after that symbol
anonymous
  • anonymous
Couldn't you just write, \[ B^{2} = (P^{-1} A P)(P^{-1} A P) \] and then use the fact that P and \(P^{-1}\) are inverses to write, \[ B^{2} = P^{-1} A (PP^{-1}) A P = P^{-1} A ( I ) A P = P^{-1} A^{2} P \] A similar arguement should get you a formula for \( B^{k} \). Once you have the formula for \( B^{k} \) you should be able to multiply that by P on the left and \( P^{-1} \) on the right you should get a formula for \( A^{k} \). Or at least you will if my quick thoughts on \( B^{k} \) are correct.

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anonymous
  • anonymous
Thanks a lot! Yes whatever B is raised to, A is raised to as well and the P's stay the same.
anonymous
  • anonymous
Now I'm just working on finding B^k and A^k
anonymous
  • anonymous
I haven't thought much more about it, but the formula for \( B^{k} \) should be just an extention of the \( B^{2} \).
anonymous
  • anonymous
I mean the "proof" of the formula.....
anonymous
  • anonymous
sorry just I haven't been great with proofs, I've got \[B ^{k}= B B ^{K-1} which -> =(P ^{-1}AP)(P ^{-1}AP)^{K-1}\] Then \[-> =(P ^{-1}A ^{2}P)^{K-1}\]...Is that sufficient proof? I don't know where to end it
anonymous
  • anonymous
.....Sorry I think the answer is just \[B ^{k}=P^{-1}A ^{K}P\]...and that is sufficient
anonymous
  • anonymous
And \[A ^{k}=P ^{-1}B ^{K}P\] correct? I'm not sure if we need more proof than that, it just says "find an analogous relationship involving B^k and A^k
anonymous
  • anonymous
Sorry, I've been having internet issues....
anonymous
  • anonymous
No problem, I am too
anonymous
  • anonymous
You've got the correct formula for \(B^{k}\). I think the "best" proof (although that is relative) is just to do something like, \[ B^{k} = B B .... B B \] where you multiply the B k times then just replace each B with \( P^{-1} A P \) and cancel each \( P P^{-1} \) as above. The problem is more one of writing this out. It's more of a "thought" proof I suppose. It's probably going to depend on what you actually need to do. If It's just find a formula then you probably don't need a lot of proofs...
anonymous
  • anonymous
okay Ill just do \[B ^{k}=BB ^{K-1}B ^{K-2}...B ^{K-K}\]
anonymous
  • anonymous
For the \(A^{k}\) I think you've got the P's backwards. Remember that you've got to be careful with "order" of multiplication. I.e. you need to multiply on the same side of the equation. So, starting with the formula for \(B^{k}\) we can do the following \[ B^{k} = P^{-1} A^{k} P \] The multiply the left side by P and the right side by \(P^{-1}\). \[ P B^{k} P^{-1} = P P^{-1} A^{k} P P^{-1} \] which gives, \[ P B^{k} P^{-1} = A^{k} \]
anonymous
  • anonymous
I'm going to be away from my computer for a while now unfortunately. I'll try to check back later, but I don't knwo when I'll get the chance....
anonymous
  • anonymous
Thanks for everything i'm good now

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