Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C...

Given,

Matrix $A$ is symmetric, $B$ is skew symmetric and $C$ is skew symmetric,

So, ${A}^{T} = A, {B}^{T} = - B, {C}^{T} = - C$

Now let $M = {A}^{13}{B}^{26} - {B}^{26}{A}^{13}$

Then, ${M}^{T} = {\left({A}^{13}{B}^{26} - {B}^{26}{A}^{13}\right)}^{T}$

$= {\left({A}^{13}{B}^{26}\right)}^{T} - {\left({B}^{26}{A}^{13}\right)}^{T}$

$= {\left({B}^{T}\right)}^{26}{\left({A}^{T}\right)}^{13} - {\left({A}^{T}\right)}^{13}{\left({B}^{T}\right)}^{26}$

$= {B}^{26}{A}^{13} - {A}^{13}{B}^{26} = - M$

Hence, $M$ is skew symmetric

Now let, $N = {A}^{26}{C}^{13} - {C}^{13}{A}^{26}$

Then, ${N}^{T} = {\left({A}^{26}{C}^{13}\right)}^{T} - {\left({C}^{13}{A}^{26}\right)}^{T}$

$= - {\left(C\right)}^{13}{\left(A\right)}^{26} + {A}^{26}{C}^{13} = N$

Hence, $N$ is symmetric.

So, only $S2$ is true.