A^k matrix singularity and (A^k)^-1 = (A^-1)^k

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Homework Help Overview

The discussion revolves around proving properties of matrix powers, specifically that if A is a nonsingular matrix, then A^k is also nonsingular for any positive integer k, and that (A^k)^-1 equals (A^-1)^k.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss starting with specific cases, such as k=2, to build a general proof. There are attempts to clarify the definitions of singular and nonsingular matrices and the implications of these properties on the proof.

Discussion Status

Some participants are exploring the idea of proving the statement for small values of k as a strategy for generalization. Others are questioning the implications of singularity and the conditions under which a matrix has an inverse, indicating a productive exploration of the topic.

Contextual Notes

There is a mention of incomplete responses and a request for clarification on specific terms, suggesting that participants are navigating through the requirements of the problem and the definitions involved.

ramtin
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Homework Statement



Let A be nonsingular. Prove That for any positive integer k , A^k is nonsingular, And (A^k)^-1 = (A^-1)^k.

Homework Equations


The Attempt at a Solution

 
Last edited:
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What have you tried so far?
 
Fill this in: a problem that requires you to prove a simple expression is true for every positive integer [tex]k[/tex] is a good candidate for m ************ ******n
 
statdad said:
Fill this in: a problem that requires you to prove a simple expression is true for every positive integer [tex]k[/tex] is a good candidate for m ************ ******n

You answer was not complete ...What are the * ?
Please somebody help me!
 
Start small. Can you prove it's true for k=2? How can you generalize the proof?
 
Office_Shredder said:
Start small. Can you prove it's true for k=2? How can you generalize the proof?

I can't prove it for 2 ,,,don't know How to generalize that:cry:
 
Start by contradiction. If A is nonsingular, then if A2 is singular what can we prove about A? Try messing around with the equation A2v = 0
 
Then what do you know? Under what conditions is a matrix "singular"? What does having an inverse mean?
 

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