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 _____.

Let the given $4 \times 4$ matrix be $A$:

Notice that the sum of the elements in each row is:

Since the row sum is constant ($3$) for every row, $\lambda = 3$ is an eigenvalue of $A$ corresponding to the eigenvector $[1, 1, 1, 1]^T$.

Alternatively, we can express $A$ as:

where $J$ is the $4 \times 4$ all-ones matrix and $I$ is the $4 \times 4$ identity matrix.

The eigenvalues of $J_{4 \times 4}$ (which has rank 1 and trace 4) are:

Therefore, the eigenvalues of $A = J - I$ are:

The largest eigenvalue of $A$ is 3.