The characteristic polynomial is given by

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

In this example we have:

$$ \begin{aligned} p(x) &= det(A - x I) = det \left( \left[ \begin{matrix}2&-1&1\\-1&1&0\\1&0&1\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}2 - \lambda &-1&1\\-1&1 - \lambda &0\\1&0&1 - \lambda \end{matrix} \right| = \\ &= (-\lambda+2) (-\lambda+1) (-\lambda+1) + (-1) \cdot 0 \cdot 1 + 1 \cdot (-1) \cdot 0 - (-\lambda+2) \cdot 0 \cdot 0 - (-1) \cdot (-1) \cdot (-\lambda+1) - 1 \cdot (-\lambda+1) \cdot 1 = \\ & = -\lambda^3+4\lambda^2-5\lambda+2 + 0 + 0 - 0 - (-\lambda+1) - (-\lambda+1) = \\ & = -\lambda^3+4\lambda^2-3\lambda \end{aligned} $$

Characteristic polynomial calculator