Symmetrical matrix determinant 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)

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 $0$.