Hello, Can you please help solve the following problem: Given two matrices A,B and AB=BA, I need to prove that for every n: . Thanks in advance, Michael
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Originally Posted by msokol89 Hello, Can you please help solve the following problem: Given two matrices A,B and AB=BA, I need to prove that for every n: . Have you tried to prove this by induction?
I suppose to technically prove this, you would use induction. But if you just write out the definition of exponentiation ( times) the result should be clear.
Originally Posted by Plato Have you tried to prove this by induction? I have tried to prove it by induction: 1) I checked for n=1: 2)I assumed for n=k it is true: 3)I checked for n=k+1: Then, after switching places and using the induction assumption , I reached: is this ok?
Last edited by msokol89; Mar 9th 2011 at 10:15 PM.
Originally Posted by msokol89 I have tried to prove it by induction: 1) I checked for n=1: 2)I assumed for n=k it is true: 3)I checked for n=k+1: Then, after switching places and using the induction assumption , I reached: is this ok? I think the last couple steps are... out of order or something. You have . Your last line is . But... this is clearly true, it doesn't say anything about.. anything.
it proves that this is true for n=k+1, therefore the entire statement is true.
Originally Posted by msokol89 it proves that this is true for n=k+1, therefore the entire statement is true. No, it doesn't. You need to prove that . The trivial observation that doesn't do t Having said that , you can conclude that . Now, what can you conclude from that?
Originally Posted by HallsofIvy No, it doesn't. You need to prove that . The trivial observation that doesn't do t Having said that , you can conclude that . Now, what can you conclude from that? ok thanks, Do you have a way of solving the problem?
Originally Posted by msokol89 Hello, Can you please help solve the following problem: Given two matrices A,B and AB=BA, I need to prove that for every n: . It looks as though you need to use induction twice if you really want to pin this result down rigorously. First, use induction to show that for every Then use that result to prove by induction on m that for every (and hence in particular when m=n).
Originally Posted by Opalg It looks as though you need to use induction twice if you really want to pin this result down rigorously. First, use induction to show that for every Then use that result to prove by induction on m that for every (and hence in particular when m=n). I understand, thanks.
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