Mathematics 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 kaleighh533: so i was bored last night so i decided to draw some doodles they actually come ou
1 minute ago 6 Replies 3 Medals Sailor: Kinda obvious when exactly I wrote this but uhhhhh... heh: I wish you were here I know You believe you're the worst walking thing On the whole planet Not to
54 minutes ago 10 Replies 0 Medals nirmala: The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top.
4 hours ago 1 Reply 0 Medals lolokrat: can someone proof my literature review? main concern is making sure im synthesizing my sources enough .
7 hours ago 10 Replies 1 Medal Grimmcake: what can you make but cant be seen?
8 hours ago 6 Replies 0 Medals Grimmcake: what type of dress can NEVER be worn?
8 hours ago 4 Replies 1 Medal Grimmcake: what has 88 keys but cannot open a door?
8 hours ago 2 Replies 1 Medal TETSXPREME: help
8 hours ago 7 Replies 4 Medals nguyenkimanhdhd: A coating that weighs at least 2.50 mg is needed to impart adequate shelf life to a pharmaceutical tablet.
12 hours ago 0 Replies 0 Medals zoey555: a) A 50.00 mL AgNo3 solution with unknown concentration reacts with excess Na2CO3.
12 hours ago 1 Reply 0 Medals