# Four questions about properties of symmetric and skew-symmetric matrices

Problem :

For 3 X 3 matrices M and N which of the following statements is (are ) false

(a) $N^TMN$ is a symmetric or skew symmetric , according as M is symmetric or skew symmetric.

(b) MN-NM is skew symmetric $\forall$ symmetric matrices M and N.

(c) MN is symmetric $\forall$ symmetric matrices M and N .

(d) (adjM)(adjN) = adj(MN) $\forall$ invertible matrices M and N.

Sol :

(a) I tried to solve it algebrically by using 2X2 matrices rather using 3X3 matrices ( I hope result will not vary due to this) and found it is false, it also depends on N as well. ( Please verify this assertion thanks..)

(b) I am getting the value of MN -NM =0 matrix , so it is a null matrix . please confirm the result then... I think it is false.

(c) It is true..

(d) For this please guide... thanks... a lot..

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You can do that for $n\times n$ matrices directly. $n=3$ does not make things any easier. Symmetric means $M^T=M$, skew-symmetric $M^T=-M$. So just take the transpose in $a,b,c$ and see what happens. Note that $c$ is false in general.For d, note that $adj(M)=(\det M)M^{-1}$ when $M$ is invertible. –  1015 Jun 21 '13 at 17:29