find an invertible matrix $P$ such that $P^T AP$ is diagonal Let $A = \left[
\begin{matrix}
  2 & -1 & 1 \\
  -1 & 3 & 0 \\
  1 & 0 & 0
\end{matrix}
\right]$.
Find an invertible matrix $P$ such that $P^T AP$ is diagonal 
my solve is 
$0 = \det\left(\left[\begin{matrix}
  \lambda & 0 & 0 \\
  0 & \lambda & 0 \\
  0 & 0 & \lambda
 \end{matrix}\right]
-
\left[\begin{matrix}
  2 & -1 &  1 \\
  -1 & 3 & 0 \\
  1 & 0 & 0
 \end{matrix}\right]
\right)
\\=
(\lambda-2)[(\lambda-3)(\lambda)-0]-(1)[(1\times\lambda)-(-1)0]+(-1)[1\times0-(-1)(\lambda-3)]
\\=\lambda^3-5\lambda^2+4\lambda+3$ 
--> I'm stuck here

I used eigen value decomposition of $A=PDP^T$, and we get:
$P = \left[
\begin{matrix}
   -0.5665  &  0.7118  & -0.4153\\
    0.8097  &  0.5744  & -0.1200\\
   -0.1531  &  0.4042  &  0.9018
\end{matrix}
\right]$.
Also, $D=P^TAP= \left[
\begin{matrix}
   3.6996  &  0  & 0\\
    0  &  1.7609  & 0\\
   0  &  0  &  -0.4605
\end{matrix}
\right]$
am I mising something?
