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[tex]The \emph{characteristic polynomial} $\chi(\lambda)$ of the

3$3 \times 3$~matrix

\[ \left( \begin{array}{ccc}

1 & -1 & -1 \\

-1 & 1 & -1 \\

-1 & -1 & 1 \end{array} \right)\]

is given by the formula

\[ \chi(\lambda) = \left| \begin{array}{ccc}

1-\lambda & -1 & -1 \\

-1 & 1-\lambda & -1 \\

-1 & -1 & 1-\lambda \end{array} \right|.\] [/tex]

Now if I do this by develloping the minors I get a cubic equation and I can't solve it without at least 30 minutes. I find it time consuming, especially during an exam.