If A is a symmetric matrix,

It is given that, $A$ is symmetric matrix.

$\therefore A^{T}=A$                    .....(1)

Now,

$\left(A^{3}\right)^{T}$

$=\left(A^{T}\right)^{3} \quad\left[\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n}, \forall n \in \mathrm{N}\right]$

$=A^{3}$                [Using (1)]

Since $\left(A^{3}\right)^{T}=A^{3}$, so the matrix $A^{3}$ is symmetric.

If $A$ is a symmetric matrix, then $A^{3}$ is a symmetric matrix