Spectrum of a constant-diagonal matrix

MediumLinear Algebra~6m

Let $A$ be the $30 \times 30$ matrix with every diagonal entry equal to $30$ and every off-diagonal entry equal to $1$ . What is the largest eigenvalue of $A$ ?

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Write $A$ as (a multiple of the identity) plus (the all-ones matrix $J$ ). What multiple of $I$ do you need?
The all-ones matrix $J$ of size $n$ has eigenvalues $n$ (once) and $0$ ( $n-1$ times). Shift each by the identity part.

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Spectrum of a constant-diagonal matrix

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