Symmetrical matrix determinant

Question

Is it true that every symmetrical matrix has a determinant non-zero?

If so how can I prove it?

Note: A symmetrical matrix that is not zero. (thanks to the commenter that pointed it out)

Answer 1

No it is not. Consider the diagonal matrix with all values on the diagonal non-zero except one which is $0$. Its determinant is clearly $0$. The fact that it is symmetric only gives you that the eigenvalues are real. However it is enough for one eigenvalue to be $0$ for you to get a $0$ determinant. However if all rows/columns are linearly independent (the matrix is full rank) then the determinant is not $0$.

Answer 2

For a counterexample take any row with all zeros, and fill in the rest of the matrix any way at all as long as it's symmetric and not the zero matrix.

Expanding along the zero row gives zero determinant.