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January 22nd, 2014, 03:33 AM  #1 
Newbie Joined: Sep 2012 Posts: 17 Thanks: 0  square of invertible matrix
Hi, I have to show for an invertible matrix A, its square AČ is also invertible. I'm pretty sure this is correct, or did I miss something? 
January 22nd, 2014, 09:19 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  Re: square of invertible matrix
We can use We now have exactly what we are looking for on the RHS and the left hand side is gauranteed to be meaningful since both exist , are nxn and certainly commute with each other and their inverses under matrix multiplication. 
January 23rd, 2014, 01:26 AM  #3 
Newbie Joined: Sep 2012 Posts: 17 Thanks: 0  Re: square of invertible matrix
That's a good idea, thank you.

January 23rd, 2014, 01:33 AM  #4  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: square of invertible matrix Quote:
This shows that the matrix is the inverse of A^2. You really need to prove this before you can start cancelling powers of matrices. Not the other way round.  
January 25th, 2014, 06:45 AM  #5 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: square of invertible matrix
Matrix is invertible if and only if its determinant is different from 0. Observe also that so if then also . This is a way to verify that is invertible if is.

January 25th, 2014, 04:43 PM  #6  
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: square of invertible matrix Quote:
is demonstrably an inverse for AB (on both sides). Also, since , it is clear the inverse of an invertible matrix is invertible. Thus we clearly have a group structure on the set of invertible matrices. As such, we can talk about the subgroup generated by the (invertible) matrix A, and for this subgroup the laws of (integer) exponents work exactly as expected, if we set: . We then have: for ANY integers k and m, as well as: (including the special case k =2, m = 1 under discussion here). So, in my humble opinion, your concern over this misuse of exponent rules here is unfounded.  
January 25th, 2014, 11:36 PM  #7  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: square of invertible matrix Quote:
The exponent rules were not being misused, but used prematurely given the original question.  

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