The characteristic polynomial is given by

$$ p(\lambda) = det(A - \lambda I) $$

In this example we have:

$$ \begin{aligned} p(x) &= det(A - x I) = det \left( \left[ \begin{matrix}2&-1&0\\1&1&1\\0&-1&-2\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}2 - \lambda &-1&0\\1&1 - \lambda &1\\0&-1&-2 - \lambda \end{matrix} \right| = \\ &= (-\lambda+2) (-\lambda+1) (-\lambda-2) + (-1) \cdot 1 \cdot 0 + 0 \cdot 1 \cdot (-1) - (-\lambda+2) \cdot 1 \cdot (-1) - 1 \cdot (-1) \cdot (-\lambda-2) - 0 \cdot (-\lambda+1) \cdot 0 = \\ & = -\lambda^3+\lambda^2+4\lambda-4 + 0 + 0 - (\lambda-2) - (\lambda+2) - 0 = \\ & = -\lambda^3+\lambda^2+2\lambda-4 \end{aligned} $$

Characteristic polynomial calculator