Proof of an identity in determinants

Click For Summary

Discussion Overview

The discussion revolves around proving the identity involving determinants, specifically the property that relates the determinant of a matrix raised to a power to the determinant of the original matrix. Participants seek clarification and proof regarding the expression ||An||=|A|n², exploring its implications and definitions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how to prove that ||An||=|A|n², noting that the property is referenced in their book without proof.
  • Another participant suggests that the identity is a special case of the general rule det(AB) = det(A) * det(B), but questions how the squaring occurs.
  • A participant expresses confusion about the term "double determinant" and seeks clarification on its meaning.
  • One participant challenges the existence of the identity, stating that the correct relationship involves the adjugate of a matrix and provides an alternative expression involving |adj(adjA)|.
  • Another participant reflects on their misunderstanding of the property and corrects their earlier assumption about the identity.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the validity of the identity in question, with some asserting its correctness while others challenge its existence and provide alternative formulations. The discussion remains unresolved.

Contextual Notes

There are limitations in the definitions and interpretations of terms such as "double determinant" and the specific conditions under which the identity may hold. The discussion also reflects varying levels of familiarity with determinant properties.

zorro
Messages
1,378
Reaction score
0
How to prove that ||An||=|A|n2?

This property is used in my book but they did not give any explanation/proof of it.
Can someone help?

Edit: n2=n2
 
Last edited:
Physics news on Phys.org
Hi Abdul! :smile:

(you can do sup within sup: An2 :wink:)

It's just a special case of the general rule detAB = detA*detB. :smile:
 
tiny-tim said:
It's just a special case of the general rule detAB = detA*detB. :smile:

here A=|An| and B=1
How does it get squared?
 
Perhaps I'm misunderstanding the question :redface: …

what did you mean by ||An|| ?
 
double determinant of An... Is there any other meaning?
 
Any idea?
 
I'm not familiar with the term "double determinant". Can you define it?
 
ehh..there is no such identity :redface:
The step is actually |adj(adjA)|=||A|n-2A|=(|A|(n-2)n)|A|.
I thought it is goes like ||A|n-2|=|A|(n-2)n, but that's wrong.
I figured out that the above property is |kA|=kn|A|, where n is the order of A.
 

Similar threads

Addition problem in Serge Lang Basic Mathematics
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Prove Even # Transpositions in Identity Permutation w/ Induction & Contradiction
  • · Replies 8 ·
Replies
8
Views
5K
Undergrad Proof for subgroup -- How prove it is a subgroup of Z^m?
  • · Replies 11 ·
Replies
11
Views
3K
High School Inductive proof for multiplicative property of sdet
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
Help with chemical potential formula in a mixture
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Can Hamson-Linde cycle liquidize N2?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Prove that F^infinity is infinite dimensional
  • · Replies 26 ·
Replies
26
Views
7K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K
Undergrad Is the identity of a group unique?
  • · Replies 2 ·
Replies
2
Views
2K