<s>[INST] Inequalities when matrix is negative definite 
Let $$A = \begin{pmatrix} x&1&y \\ 1&-1&0 \\ y&0&-2 \end{pmatrix} $$ $A$ is negative definite $\iff x<a \text{ and } bx^2 + cy^2 + dx + 2 < 0$. Find $a,b,c,d$


I know that for a matrix to be negative definite all of its eigenvalues must be negative (I thought this may link to the $<0$ of the equation from the question)
I have attempted to form an equation for the eigenvalues of A, by forming the characteristic polynomial as follows:
$$|A-\lambda*I_3|=
\begin{vmatrix} x-\lambda&1&y \\ 1&-1-\lambda&0 \\ y&0&-2-\lambda\end{vmatrix}$$
$$ =x-\lambda \begin{vmatrix} -1-\lambda&0 \\ 0&-2-\lambda \end{vmatrix} -1 \begin{vmatrix} 1&0 \\ y&-2-\lambda \end{vmatrix}+y\begin{vmatrix} 1&-1-\lambda \\ y&0 \end{vmatrix}$$
$$=\lambda^2x+3\lambda x+2x-\lambda^3-3\lambda^2-2\lambda+2+\lambda-y^2-\lambda y^2$$
I am then unsure what to do with this to find the answer, I tried factoring out $x$ and $y$ but that did not seem to help?
 [/INST] $A$ is negative definite iff $(-A)$ is positive definite.  To test the positive definiteness of $(-A)$, we have to calculate all three principal determinants of $(-A)$
$-A = \begin{bmatrix} -x && - 1 && - y \\ -1 && 1 && 0 \\ - y && 0 && 2 \end{bmatrix}$
$\Delta_1 = - x \gt 0   \Longleftrightarrow x \lt 0$
$\Delta_2 = -x - 1 \gt 0 \Longleftrightarrow x \lt -1 $
$\Delta_3 = 2 (-x - 1) - y (y) = -2 x - y^2 - 2 \gt 0 \Longleftrightarrow y^2 + 2 x + 2 \lt 0 $
Hence, $a = -1, b = 0, c = 1, d = 2 $
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