Consider the following matrix: \[ \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{pmatrix} \] The largest eigenvalue of the above matrix is \(\underline{\hspace{2cm}}\).
Consider the following matrix: \[ \begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{pmatrix} \] The largest eigenvalue of the above matrix is \(\underline{\hspace{2cm}}\).
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Correct Answer: 3
Solution and Explanation
For a \( 4 \times 4 \) matrix:
- The eigenvalues of \( J \) are \( 4 \) and \( 0 \) (with multiplicity 3).
- Subtracting \( I \) reduces each eigenvalue by 1.
Thus, eigenvalues of \( A \) are: \[ 4 - 1 = 3, 0 - 1 = -1 \] Hence, the largest eigenvalue is: \[ 3 \] Final Answer: \[ \boxed{3} \]
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