Mathematics 40 Online OpenStudy (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 OpenStudy (anonymous):

im new plz tell me what does ^ means OpenStudy (anonymous):

It means "to the exponent of" so whatever comes before that symbol is raised to the exponent of whatever comes after that symbol OpenStudy (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. OpenStudy (anonymous):

Thanks a lot! Yes whatever B is raised to, A is raised to as well and the P's stay the same. OpenStudy (anonymous):

Now I'm just working on finding B^k and A^k OpenStudy (anonymous):

I haven't thought much more about it, but the formula for \( B^{k} \) should be just an extention of the \( B^{2} \). OpenStudy (anonymous):

I mean the "proof" of the formula..... OpenStudy (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 OpenStudy (anonymous):

.....Sorry I think the answer is just \[B ^{k}=P^{-1}A ^{K}P\]...and that is sufficient OpenStudy (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 OpenStudy (anonymous):

Sorry, I've been having internet issues.... OpenStudy (anonymous):

No problem, I am too OpenStudy (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... OpenStudy (anonymous):

okay Ill just do \[B ^{k}=BB ^{K-1}B ^{K-2}...B ^{K-K}\] OpenStudy (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, \[ P B^{k} P^{-1} = A^{k} \] OpenStudy (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.... OpenStudy (anonymous):

Thanks for everything i'm good now

Latest Questions rose12345: what are three ways to keep the chicks safe from hawks and weather
2 hours ago 1 Reply 0 Medals kewss: An astronaut pushes a button on his control panel and a light turns on.
13 hours ago 0 Replies 0 Medals Rango: any sites to research urban legends?
15 hours ago 9 Replies 1 Medal Aries: Help Please!
16 hours ago 1 Reply 0 Medals GhostlyEnigma: Simplify the following expression: 4x + 6y - 3x + 8y - 2x - 5y
18 hours ago 3 Replies 3 Medals virgo1234: Perform the following calculation of measured numbers. Give the answer with the correct number of significant figures.
18 hours ago 2 Replies 1 Medal Vixen: Need help on chem ud83eudd74
15 hours ago 8 Replies 3 Medals Kimberly44: what yall think!!?
15 hours ago 8 Replies 4 Medals Extrinix: So I found a few, well, interesting themes from back then. ud83duddff Oh well
1 day ago 5 Replies 3 Medals Alanaaaaaaa: Love is a force that moves us all, A flame that burns within, It lifts us up and makes us whole, And fills our hearts with kin.
15 hours ago 5 Replies 4 Medals