Which of the following quadratic forms defines a non-singular conic?
(1). $x_{0}^{2}-2x_{0}x_{1}+4x_{0}x_{2}-8x_{1}^{2}+2x_{1}x_{2}+4x_{2}^{2}$.
(2). $x_{0}^{2}-4x_{0}x_{1}+x_{1}^{2}-2x_{0}x_{2}$.
What is a good way to solve this problem? Thanks a lot.
The matrix for the first conic is $$ \begin{bmatrix} 1 & -1 & 2 \\ -1 & -8 & 1 \\ 2 & 1 & 4 \end{bmatrix} $$ which has determinant $-9$.
The matrix for the second conic is $$ \begin{bmatrix} 1 & -2 & -1 \\ -2 & 1 & 0 \\ -1 & 0 & 0 \end{bmatrix} $$ which has determinant $-1$.
So neither is singular.