A = \(\begin{bmatrix}1 & 2 & 1 & 0\\1 & 0 & 1 & 0\\2 & 1 & 1 & 0\\3 & 1 & 0 & 1\end{bmatrix}\)
\(P_{A}\)(\(x\)) = \(det(\)A - \(x\) I\(_4\)) = \(det(\begin{bmatrix}1 & 2 & 1 & 0\\1 & 0 & 1 & 0\\2 & 1 & 1 & 0\\3 & 1 & 0 & 1\end{bmatrix}\) - \(x\)\(\begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix})\)
= \(det\begin{bmatrix}1-x & 2 & 1 & 0\\1 & -x & 1 & 0\\2 & 1 & 1-x & 0\\3 & 1 & 0 & 1-x\end{bmatrix}\)
= (1-\(x\))\(\begin{bmatrix}1-x & 2 & 1 \\1 & -x & 1 \\2 & 1 & 1-x\end{bmatrix}\)
= (1-\(x\))[\(\begin{bmatrix}1 & -x\\2 & 1\end{bmatrix}\) - \(\begin{bmatrix}1-x & 2\\2 & 1\end{bmatrix}\) + (1-\(x\))\(\begin{bmatrix}1-x & 2\\1 & -x\end{bmatrix}\)]
= (1-\(x\))[(1+2\(x\))-(1-\(x\)-4)+(1-\(x\))(-\(x\)+\(x^2\)-2)]
= (1-\(x\))(\(-x^3+2x^2+4x+2\))
= \(x^4-3x^3-2x^2+2x+2\)