Consider the following matrix. pmatrix 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 pmatrix The largest eigenvalue of the above matrix is ________.

GATE CSE 2021 Set 1 Q62: Largest eigenvalue of J − I for a 4×4 all-ones matrix

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 ________.

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Answer: 3 (official GATE 2021 final answer key)

Concept used: The matrix is $J - I$ , where $J$ is the all-ones matrix with eigenvalues 4, 0, 0, 0.

First step: Multiply the matrix by the all-ones vector: every row sums to 3.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key