  Positivity properties of the Hadamard powers of the matrix
$\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers
$x_1,\ldots,x_n$ and the matrix
$\begin{bmatrix}|\cos((i-j)\pi/n)|\end{bmatrix}$ are studied. In particular, it
is shown that $\begin{bmatrix}(1+x_ix_j)^r\end{bmatrix}$ is not positive
semidefinite for any positive real number $r<n-2$ that is not an integer, and
$\begin{bmatrix}|\cos((i-j)\pi/n)|^r\end{bmatrix}$ is positive semidefinite for
every odd integer $n\ge 3$ and $n-3\le r<n-2.$
