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Suppose a square matrix $A$ has non-negative entries and each row sum is at least $a$. Does it mean that the spectral radius of $A$ is at least $a$?

If $A$ has all positive entries or is irreducible in the Perron-Frobenius sense, then this is clearly true. What if it's not?

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Hint: once you have proved it for the irreducible case the reducible case follows by a continuity argument. You can consult, e.g., Minc, Nonnegative matrices

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