A and B are two symmetric matrices

In summary, A and B are two symmetric matrices that satisfy the condition AB = - BA. From this, it can be deduced that (A-B)^2 is always symmetric. However, the statement "AB^2 is symmetric" is not always true, as shown by the example provided. Additionally, the invertibility of AB cannot be determined from the given information.
  • #1
Yankel
395
0
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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  • #2
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
 
  • #3
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 ?
 
  • #4
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:
  • #5
the 2nd can't be true. I just found an example...solved, thanks !
 

FAQ: A and B are two symmetric matrices

What is the definition of symmetric matrices A and B?

Symmetric matrices A and B are square matrices where the elements are symmetric across the main diagonal. This means that for any given element at position (i,j), the element at position (j,i) must be the same value.

How can you determine if a matrix is symmetric?

To determine if a matrix is symmetric, you can compare the transpose of the matrix to the original matrix. If they are the same, then the matrix is symmetric. Alternatively, you can also check if the elements are symmetric across the main diagonal.

Can a matrix be both symmetric and skew-symmetric?

No, a matrix cannot be both symmetric and skew-symmetric. A symmetric matrix has symmetric elements across the main diagonal, while a skew-symmetric matrix has elements that are equal in magnitude but opposite in sign across the main diagonal.

What are the properties of symmetric matrices A and B?

Some properties of symmetric matrices A and B include: they are always square matrices, their transpose is equal to the original matrix, and their eigenvalues are all real numbers.

How are symmetric matrices used in scientific applications?

Symmetric matrices have many applications in science, such as in physics, engineering, and statistics. For example, they are used in quantum mechanics to represent operators, in structural engineering to analyze symmetric structures, and in statistical analysis to model symmetric data.

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