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.