Given a symmetric matrix $n\times n$
$$A_{i,j} = \begin{cases} \alpha & \text{ if } i=j\\ 1 & \text{ if } i \neq j \end{cases}$$
Prove that $A$ is positive semidefinite iff $\alpha \geq 1$.
I managed to show that by assuming $\alpha < 1$ we get a contradiction that A is positive semidefinite but the second side I thought maybe using Sylvester's criterion?