Question

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 \(\underline{\hspace{2cm}}\).

Hint:

Matrices of the form \( J - I \) have one large eigenvalue and remaining identical smaller eigenvalues.

Answer 1

Correct Answer: 3

The given matrix can be written as: \[ A = J - I \] where \( J \) is a matrix of all 1's and \( I \) is the identity matrix.
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: \[ \boxed{3} \]