Calculating eigenvalues of a matrix I have this quadratic matrix:
\begin{align*}
A = \begin{bmatrix}
0 & -1 & -2\\
-1 & 0 & -2\\
-2 & -2 & -3
\end{bmatrix} \implies A - \lambda I = \begin{bmatrix}
0 - \lambda & -1 & -2\\
-1 & 0 - \lambda & -2\\
-2 & -2 & -3 - \lambda
\end{bmatrix}
\end{align*}
After forming the characteristic polynomial and using the Rule of Sarrus, I get this equation:
$$-\lambda^3 - 3\lambda^2 + 9\lambda - 5$$
However the correct form of this term would be:
$$-(5 + \lambda)(1 - \lambda)^2$$
So the eigenvalues are $1$ and $-5$.
The thing I do not understand, is how to form this equation to get the values. Which approach and rules can solve this?
Thank you very much.


$$p(\lambda)=-\lambda^3 - 3\lambda^2 + 9\lambda - 5$$

Proceed from your work,
$$\begin{align}
p(\lambda)&=-\lambda^3-5\lambda^2+2\lambda^2+9\lambda-5\\
\\
&=-\lambda^2(\lambda+5)+(2\lambda-1)(\lambda+5)\\
\\
&=(\lambda+5)(-\lambda^2+2\lambda-1)\\
\\
&=-(\lambda+5)(\lambda^2-2\lambda+1)\\
\\
&=-(\lambda+5)(\lambda-1)^2\end{align}
$$
