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Manu
Average Member

Australia
9 Posts
Posted - 03/27/2012 :  10:03:11  Show Profile  Reply with Quote
For each of the following statements, select whether the statement is true or false for all n × n invertible matrices A, B , C. (Note that you are being asked whether the statement is true or false for all n × n invertible matrices A, B, C, not just for some invertible A, B, C.)

1. (ABA^-1)^8 = A^8B^8A^-8

2. B^12 = B^4 ⇒ B^-1 = B^7

3. A^-1 = BC ⇒ C^-1 = AB

Is the answer 1-true, 2-false, 3-true is correct.
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royhaas
Moderator

USA
3044 Posts
Posted - 03/28/2012 :  08:50:50  Show Profile  Reply with Quote
Why do you think 2 is false?
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Manu
Average Member

Australia
9 Posts
Posted - 03/28/2012 :  10:08:21  Show Profile  Reply with Quote
I think B^12 = B^4 is not possible.
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royhaas
Moderator

USA
3044 Posts
Posted - 03/28/2012 :  12:25:14  Show Profile  Reply with Quote
quote:
Originally posted by Manu

I think B^12 = B^4 is not possible.


It is true if B=I. Also, consider the 2X2 complex matrix diag[(1+i)/2]. It's easy to show that B^8=I. There are other examples.
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