  The problem of expressing a multivariate polynomial as the determinant of a
monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has
drawn a huge amount of attention due to its connection with optimization
problems. In this paper we provide a necessary and sufficient condition for the
existence of \textit{monic Hermitian determinantal representation} as well as
\textit{monic symmetric determinantal representation} of size $2$ for a given
quadratic polynomial. Further we propose a method to construct such a monic
determinantal representtaion (MDR) of size $2$ if it exists. It is known that a
quadratic polynomial $f(\x)=\x^{T}A\x+b^{T}\x+1$ has a symmetric MDR of size
$n+1$ if $A$ is \textit{negative semidefinite}. We prove that if a quadratic
polynomial $f(\x)$ with $A$ which is not negative semidefinite has an MDR of
size greater than $2$, then it has an MDR of size $2$ too. We also characterize
quadratic polynomials which exhibit diagonal MDRs.
