When is this matrix positive definite?

Question

A square matrix $M \in \mathbb{R}^{n \times n}$ is defined by

$$M_{i,j} = \begin{cases} 1 & \text{if } i=j \\ p & \text{otherwise}\end{cases}$$

where $M_{i,j}$ denotes the $(i,j)$-th entry. What is the "iff" condition for $p$ to make the matrix $M$ positive definite?

Answer 1

Given a matrix $M_{n\times n}$ defined as proposed, its eigenvalues are

$$ \{1-p,\cdots,1-p,1+(n-1)p\} $$ so the conditions are

$$ \cases{ 1-p > 0\\ 1+(n-1)p > 0 } $$

or

$$ -\frac{1}{n-1}< p < 1 $$