For \(n \ge 2\) the determinant of an \(n \times n\) matrix \(A = [{a_{ij}}]\) is the sum of \(n\)terms of the form \( \pm {a_{1j}}\det {A_{1j}},\) with plus and minus signs alternating, where the entries \({a_{11}},{a_{12}}, \ldots ,{a_{1n}}\) are from the first row of \(A\). In symbols,
\(\begin{gathered} {\text{det}}A = {a_{11}}\det {A_{11}} - {a_{12}}\det {A_{12}} + \ldots + {\left( { - 1} \right)^{1 + n}}{a_{1n}}\det {A_{1n}} \\ = \mathop \sum \limits_{j = 1}^n {\left( { - 1} \right)^{1 + j}}{a_{1j}}\det {A_{1j}} \\ \end{gathered} \)
With the help of above defined formula, the \(\det A\) is calculated as follows:
\(\begin{array}{c}det\left( {A - \lambda I} \right) = det\left( {\begin{array}{*{20}{c}}{ - \lambda }&3&1\\3&{ - \lambda }&2\\1&2&{ - \lambda }\end{array}} \right)\\ = \left( { - \lambda } \right)\left| {\begin{array}{*{20}{c}}{ - \lambda }&2\\2&{ - \lambda }\end{array}} \right| - 3\left| {\begin{array}{*{20}{c}}3&2\\1&{ - \lambda }\end{array}} \right| + 1\left| {\begin{array}{*{20}{c}}3&{ - \lambda }\\1&2\end{array}} \right|\\ = \left( { - \lambda } \right)\left( {\left( { - \lambda } \right)\left( { - \lambda } \right) - \left( 2 \right)\left( 2 \right)} \right) - 3\left( {\left( 3 \right)\left( { - \lambda } \right) - \left( 2 \right)\left( 1 \right)} \right) + 1\left( {\left( 2 \right)\left( 3 \right) - \left( { - \lambda } \right)\left( 1 \right)} \right)\\ = 4\lambda - {\lambda ^3} + 6 + 9\lambda + 6 + \lambda \\ = - {\lambda ^3} + 14\lambda + 12\end{array}\)
Thus, the characteristic polynomial is \( - {\lambda ^3} + 14\lambda + 12\).
