Given that AT=A,BT=−B
Now solving all option one by one we get,
Solving (A) option
C=A4−B4
CT=(A4−B4)=(A4)T−(B4)T=A4−B4=C
⇒C=CT is a symmetric matrix
Solving (B) option
C=AB−BA
CT=(AB−BA)T=(AB)T−(BA)T
=BTAT−ATBT=−BA+AB=C
⇒C=CT is a symmetric matrix
Solving (C) option
C=B5−A5
CT=(B5−A5)T=(B5)T−(A5)T=−B5−A5
⇒C=CTorC=−CT
Solving (D) option
C=AB+BA
CT=(AB+BA)T=(AB)T+(BA)T
=−BA−AB=−C
C=−CT is skew symmetric matrix
∴ Option (C) is not true.