Leading Eigenvalue of the Complete Graph

Consider the complete graph on \(n\) vertices, denoted \(K_n\). This is the simple graph (without self-loops or multi-edges) with \(n\) vertices and an edge between every pair of vertices.

We also often refer to this graph as an \(n\)-clique.

Find the largest eigenvalue of the adjacency matrix of this graph, and explain your solution.

Hint: Express \(\mathbf{A}\) in relation to the matrix \(\mathbf{E} \in \mathbb{R}^{n\times n}\) which contains 1 in every entry. You may use without proof the fact that \(\mathbf{E}\) has eigenvalue \(n\) with multiplicity 1 and eigenvalue 0 with multiplicity \(n-1\).