All positive integral powers of a symmetric matrix are:

The correct answer is Option (3) → Symmetric

Let A be symmetric, so \(A^T = A\). Using the reversal law \((AB)^T = B^T A^T\) repeatedly, for any positive integer n,

    \((A^n)^T = (A^T)^n\)
    \(= A^n\)

So every positive integral power \(A^n\) is symmetric. A symmetric matrix such as \(\begin{bmatrix} 1 & 2\\ 2 & 1\end{bmatrix}\) has powers that are neither diagonal, skew-symmetric nor the identity.