Find minimal a for PSD of 3x3 matrix

Determine the smallest real value of $a$ for which the symmetric matrix whose diagonal entries are all 1 and whose off-diagonal entries are all $a$,

$A = \begin{bmatrix}1 & a & a\\ a & 1 & a\\ a & a & 1\end{bmatrix},$

is positive semidefinite. Justify your answer using eigenvalues (or Sylvester’s criterion), showing every step.

Overview: This question tests understanding of linear algebra ideas such as eigenvalues, matrix definiteness, and spectral properties by asking for the smallest real parameter $a$ that makes a symmetric $3 \times 3$ matrix with unit diagonal and constant off-diagonal entries positive semidefinite.