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Consider the matrix \(\mathbf{B} = \begin{pmatrix} 3 & -1 \\ -1 & 3 \end{pmatrix}\) and the vector \(\mathbf{u} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\).
  1. Calculate the product \(\mathbf{B}\mathbf{u}\).
  2. Hence, determine the eigenvalue \(\lambda\) associated with the eigenvector \(\mathbf{u}\).

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