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May 7th, 2013, 06:07 AM   #1
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With characteristic polynomial show that A is invertible

Show that square matrix [itex]A[/latex] is reversible if [itex]p_{A}(0)\neq 0[/latex] where [itex]p_{A}[/latex] is characteristic polynomial. Prove similar statement for minimal polynomial.

The first part should be very similar to this (if not completely correct?):
If [itex]A[/latex] is reversible than [itex]detA\neq 0[/latex].
Characteristic polynomial is by it's definition calculated as .
for than:
but is reversible and therefore so .

End of part one (the other direction is similar).

Part two: Minimal polynomial:

For minimal polynomial but I am not sure how to continue.. BTW, why would this differ from the first part at all? could easily also be minimal?
If in characteristic polynomial I insert instead of : but I don't see how this proves that is reversible. Could somebody give ma hint what's the whole idea here? :/

Thanks for all the help!

This is actually Home Work I have to do so if there is a separated topic for this kind of questions I sincerely apologize!
skrat is offline  
 
May 7th, 2013, 06:09 AM   #2
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Re: With characteristic polynomial show that A is invertible

SORRY for the mistakes in first post:
Quote:
Originally Posted by skrat
Show that square matrix is reversible if where is characteristic polynomial. Prove similar statement for minimal polynomial.

The first part should be very similar to this (if not completely correct?):
If is reversible than .
Characteristic polynomial is by it's definition calculated as .
for than:
but is reversible and therefore so .

End of part one (the other direction is similar).

Part two: Minimal polynomial:

For minimal polynomial but I am not sure how to continue.. BTW, why would this differ from the first part at all? could easily also be minimal?
If in characteristic polynomial I insert instead of : but I don't see how this proves that is reversible. Could somebody give ma hint what's the whole idea here? :/

Thanks for all the help!

This is actually Home Work I have to do so if there is a separated topic for this kind of questions I sincerely apologize!
skrat is offline  
May 17th, 2013, 04:04 AM   #3
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Re: With characteristic polynomial show that A is invertible

Don't insert "A" or (that doesn't really make sense, is a number, not a matrix), insert . In that case becomes , exactly what you want.
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