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 expressed as: \[ A = J - I \] where: - \( J \) is the \(4 \times 4\) matrix with all entries equal to 1, - \( I \) is the \(4 \times 4\) identity matrix.

Step 1: Eigenvalues of \( J \)
For an \( n \times n \) all-ones matrix \( J \): - One eigenvalue is \( n \), - The remaining \( n-1 \) eigenvalues are \( 0 \). Thus, for \( n = 4 \), the eigenvalues of \( J \) are: \[ 4,\; 0,\; 0,\; 0 \]

Step 2: Effect of subtracting the identity matrix
Subtracting the identity matrix \( I \) from \( J \) reduces each eigenvalue by 1. Therefore, the eigenvalues of \( A = J - I \) are: \[ 4 - 1 = 3,\quad 0 - 1 = -1 \; (\text{with multiplicity } 3) \]

Step 3: Largest eigenvalue
Among the eigenvalues \( \{3, -1, -1, -1\} \), the largest eigenvalue is: \[ 3 \]

Final Answer:
\[ \boxed{3} \]