Let A and B be any two 3× 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Given that ${A}^{T}=A,{B}^{T}=-B$AT=A,BT=−B

Now solving all option one by one we get,

Solving $(A)$(A) option

$$C={A}^{4}-{B}^{4}$$C=A4−B4

$${C}^{T}=({A}^{4}-{B}^{4})={({A}^{4})}^{T}-{({B}^{4})}^{T}={A}^{4}-{B}^{4}=C$$CT=(A4−B4)=(A4)T−(B4)T=A4−B4=C

$\Rightarrow C={C}^{T}$⇒C=CT is a symmetric matrix

Solving $(B)$(B) option

$$C=AB-BA$$C=AB−BA

$${C}^{T}={(AB-BA)}^{T}={(AB)}^{T}-{(BA)}^{T}$$CT=(AB−BA)T=(AB)T−(BA)T

$$={B}^{T}{A}^{T}-{A}^{T}{B}^{T}=-BA+AB=C$$=BTAT−ATBT=−BA+AB=C

$\Rightarrow C={C}^{T}$⇒C=CT is a symmetric matrix

Solving $(C)$(C) option

$$C={B}^{5}-{A}^{5}$$C=B5−A5

$${C}^{T}={({B}^{5}-{A}^{5})}^{T}={({B}^{5})}^{T}-{({A}^{5})}^{T}=-{B}^{5}-{A}^{5}$$CT=(B5−A5)T=(B5)T−(A5)T=−B5−A5

$$\Rightarrow C\neq {C}^{T}\text{or}C\neq -{C}^{T}$$⇒C=CTorC=−CT

Solving $(D)$(D) option

$$C=AB+BA$$C=AB+BA

$${C}^{T}={(AB+BA)}^{T}={(AB)}^{T}+{(BA)}^{T}$$CT=(AB+BA)T=(AB)T+(BA)T

$$=-BA-AB=-C$$=−BA−AB=−C

$C=-{C}^{T}$C=−CT is skew symmetric matrix

$\therefore$∴ Option (C) is not true.