Eigenvalues of different symmetric $(2n+1)\times(2n+1)$ matrix 
I ve looked at other similar post but I could not find help with them

Write down $det(A-xI)$ and you do basic determinant operation then you will find $characteristics~polynomial$ from there you can find eigen values.
Hint: The characteristics polynomial will be
$det(A-xI)=x^{2n+1}-2nx^{2n-1}$
From here you will find that eigen values of this matrix are $ 0~and~\pm \sqrt{2n}$
