Show that the \(3 \times 3\) Hermitian matrix

\[
\left(\begin{array}{lll}
1 & ho & \bar{ho} \\
\bar{ho} & 1 & ho \\
ho & \bar{ho} & 1
\end{array}ight)
\]

has determinant \(1-3|ho|^{2}+2 \mathfrak{R}\left(ho^{3}ight)\). Hence or otherwise, deduce that the matrix is positive definite if and only if \(ho\) lies in the triangle with vertices at the cube roots of unity \(\left\{1, e^{2 \pi i / 3}, e^{-2 \pi i / 3}ight\}\).