A and B are two symmetric matrices

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Discussion Overview

The discussion revolves around properties of two symmetric matrices A and B that satisfy the condition AB = -BA. Participants explore which of three statements regarding these matrices are always true, focusing on the symmetry of certain expressions and the invertibility of the product AB.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that (A-B)^2 is symmetric, citing the expansion A^2 - AB - BA + B^2 and noting that A^2 and B^2 are symmetric.
  • Others express uncertainty about the implications of the condition AB = -BA and its relevance to the other statements.
  • One participant claims that the second statement, AB^2 being symmetric, cannot be true and mentions finding an example that supports this conclusion.

Areas of Agreement / Disagreement

Participants generally agree on the symmetry of (A-B)^2, but there is disagreement regarding the truth of the other two statements, particularly the symmetry of AB^2 and the invertibility of AB. The discussion remains unresolved regarding the latter two statements.

Contextual Notes

Participants express limitations in understanding the implications of the condition AB = -BA and seek examples to clarify the properties of the matrices involved.

Yankel
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A and B are two symmetric matrices that satisfy: AB = - BA

Which one of these statements are always true:

a. (A-B)^2 is symmetric

b. AB^2 is symmetric

c. AB is invertable

I tried to think of an example for such matrices, but couldn't even find 1...there must be a logical way to solve it.

any assistance will be appreciated...
 
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Yankel said:
A and B are two symmetric matrices that satisfy: AB = - BA

Which one of these statements are always true:

a. (A-B)^2 is symmetric

b. AB^2 is symmetric

c. AB is invertable

I tried to think of an example for such matrices, but couldn't even find 1...there must be a logical way to solve it.

any assistance will be appreciated...

Consider (a):

Expand (A-B)^2=(A-B)(A-B)=A^2-AB-BA+B^2=A^2+B^2

If a matrix U is symmetric then so is U^2 so ..

CB
 
right, so if A^2 is symmetric and B^2, so A^2 + B^2 must be...thanks for that.

any idea about any of the other two ? I don't get the AB=-BA condition, what does it give me ?
 
Yankel said:
any idea about any of the other two ? I don't get the AB=-BA condition, what does it give me ?

The problem asks "Which one of these statements are always true?" So...
 
Last edited:
the 2nd can't be true. I just found an example...solved, thanks !
 

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