MHB A and B are two symmetric matrices

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A and B are symmetric matrices that satisfy the condition AB = -BA. It is established that (A-B)^2 is symmetric, as the expansion leads to A^2 + B^2, both of which are symmetric. However, the second statement, AB^2 being symmetric, is not always true, as demonstrated by a counterexample. The invertibility of AB is also questioned, with no definitive conclusion reached in the discussion. The participants acknowledge the complexity of the problem and express gratitude for insights shared.
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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