Welcome to our community

Be a part of something great, join today!

A and B are two symmetric matrices

Yankel

Active member
Jan 27, 2012
398
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...
 

CaptainBlack

Well-known member
Jan 26, 2012
890
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
 

Yankel

Active member
Jan 27, 2012
398
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 ?
 

Alexmahone

Active member
Jan 26, 2012
268
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:

Yankel

Active member
Jan 27, 2012
398
the 2nd can't be true. I just found an example...solved, thanks !