If A, B symmetric; show AB+BA symm., AB-BA skew symm.

[jas]

if A and B are symmetric matrices of same order then show that 1. AB+BA is symmetric 2. AB-BA is skew symmetric

 

[Deleted member 4993]

Re: matrix

these answers are not in my textbook pls. if you know its answers then pls. tell me also.

Answers of what???

 

[masi]

if A and B are symmetric matrices of same order then show that 1. AB+BA is symmetric 2. AB-BA is skew symmetric

These are both immediate results of the basic properties of the transpose operator. Without just giving you the answers, consider using these properties which hold for all \(\displaystyle A,B \in M_n(F)\) (i.e. matrices with the same order):

(i) \(\displaystyle (A+B)^T = A^T + B^T\)

(ii) \(\displaystyle (AB)^T = B^T A^T\)

(iii) \(\displaystyle A\) is said to be symmetric if \(\displaystyle A^T = A\)

(iv) \(\displaystyle A\) is said to be skew-symmetric if \(\displaystyle A^T = -A\)

So in your work, use these properties to obtain \(\displaystyle (AB+BA)^T\) and \(\displaystyle (AB-BA)^T\) and observe that (iii) and (iv) are met as desired.