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Let \(J = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \in M_3(\mathbb{R})\). Show that \(J^2 = 3 J\), and deduce \(J^k\) for every \(k \ge 1\).
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