Find the characteristic polynomial of the matrix A= \(\begin{bmatrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ \end{bmatrix}\)
\(pA(x) = det \left( \begin{bmatrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \\ \end{bmatrix} - \begin{bmatrix} x & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & x \\ \end{bmatrix} \right)\)
\(= det \left(\begin{bmatrix} 3-x & 2 & 1 \\ 0 & 1-x & 1 \\ 1 & 2 & -x \\ \end{bmatrix} \right)\)
\(= 3-x\cdot\begin{vmatrix} 1-x & 1 \\ 2 & -x \\ \end{vmatrix} - 2\cdot\begin{vmatrix} 0 & 1 \\ 1 & -x \\ \end{vmatrix} + 1\cdot\begin{vmatrix} 0 & 1-x \\ 1 & 2 \\ \end{vmatrix}\)
\(= (3-x)(-x+x^2-2)-2(-1) + x-1\)
\(= -3x +x^2+3x^2-x^3-6+2x+2+x-1\)
\(-x^3+4x^2+0-5\)
The characteristic polynomial is: \(-x^3+4x^2-5\)