Are there simple methods for calculating the determinant of symmetric matrices? I've seen that there are lots of exercises about determinants of symmetric matrices in my algebra books. Some are easy and others are a bit more twisted, but the basic problem is almost always the same. I have been trying to come up with a method to calculate these a bit more quickly, since—at least for me—they invariably end with a very ugly stream of numbers and letters.
For example I started with a $3\times 3$ matrix like this:
$$A= \begin{pmatrix}
a & b & c \\
b & a & b \\
c & b & a \end{pmatrix}$$
which looks fairly simple, but the best I could come up with for the determinant was:
$$2b^2(c-a)+a(a^2-c^2)
\quad
\text{ or }
\quad
a(a^2-2b^2-c^2)+2b^2c$$
These look horrific and absolutely not what anyone in his right mind would use. It goes without saying that I haven't even tried this with matrices bigger than $3\times 3$. Is there something I have been missing, or is there nothing to do about it?

I would rather write:
$$(a-c)(a^2 + ac - 2b^2)$$
which clearly reflects that the determinant is $0$ if $a = c$ (because two rows would be equal). You should try in higher dimensions to see if a pattern emerges and/or search whether some properties of symmetric matrices can be projected into the determinant's expression.
