I'm studying a strange example; the goal is to find the $c \in \mathbb{R}$ such that the real symmetric matrix $A$ is positive semidefinite.
First method To do so, is it correct to impose the determinant is non negative?
$$ A = \begin{bmatrix} 1 & c \\ c & 1 \end{bmatrix} $$
$$ 0 \leq det(A) = 1 - c^2 \quad \rightarrow \quad -1 \leq c \leq 1 $$
Second method If I apply the principal minors criterion, I should consider the two minors of $A$ and if they are non negative, the matrix is positive semidefinite.
There are two principal minors $det(1)= 1 > 0$ (deleting second row and column) and $det(1)= 1 > 0$ (deleting first row and column). So in this case $A$ is positive semidefinite (actually is positive definite) for all $c$.
The two methods give different results.