Condition on $a$ for matrix to be positive definite

Question

Assume you are given an $n\times n$ matrix with all elements equal to $a$, except for the diagonal values which are all $1$. What would be the condition on $a$ so that the matrix be positive definite?

Answer 1

Hint: there is an equivalent condition for a matrix to be positive definite. Namely, all upper left minors need to have a positive determinant. Check where this takes you.

Answer 2

This matrix is $(1-a) I + aJ$ where $J$ is the matrix of all ones. What are the eigenvalues of this matrix?

Hints:

$J$ is a rank one matrix whose only nonzero eigenvalue is $n$.

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The eigenvalues of $(1-a)I + aJ$ are $(1-a) + an$ and $1-a$. You want both of these to be positive.