# A and B are two symmetric matrices

#### Yankel

##### Active member
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
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

• Alexmahone

#### Yankel

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