Let A and B be any two 3 × 3 symmetric and skew symmetric matrices...

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

Now solving all option one by one we get,

Solving \left(A\right)$option </p><p>$C = {A}^{4} - {B}^{4}$</p><p>${C}^{T} = \left({A}^{4} - {B}^{4}\right) = {\left({A}^{4}\right)}^{T} - {\left({B}^{4}\right)}^{T} = {A}^{4} - {B}^{4} = C$</p><p>$⇒ C = {C}^{T} is a symmetric matrix

Solving \left(B\right)$option </p><p>$C = AB - BA$</p><p>${C}^{T} = {\left(AB - BA\right)}^{T} = {\left(AB\right)}^{T} - {\left(BA\right)}^{T}$</p><p>$= {B}^{T}{A}^{T} - {A}^{T}{B}^{T} = - BA + AB = C$</p><p>$⇒ C = {C}^{T} is a symmetric matrix

Solving \left(C\right)$option </p><p>$C = {B}^{5} - {A}^{5}$</p><p>${C}^{T} = {\left({B}^{5} - {A}^{5}\right)}^{T} = {\left({B}^{5}\right)}^{T} - {\left({A}^{5}\right)}^{T} = - {B}^{5} - {A}^{5}$</p><p>$⇒ C \ne {C}^{T}\text{or}C \ne - {C}^{T}$</p><p>Solving \left(D\right)$ option

$C = AB + BA$

${C}^{T} = {\left(AB + BA\right)}^{T} = {\left(AB\right)}^{T} + {\left(BA\right)}^{T}$

$= - BA - AB = - C$

$C = - {C}^{T} is skew symmetric matrix</p><p>$∴ Option (C) is not true.