  The underlying mixing of quark components of scalar mesons is probed in $\pi
K$ scattering within a generalized linear sigma model that contains two scalar
meson nonets and two pseudoscalar meson nonets (a quark-antiquark and a
four-quark). In the leading order of this model, all free parameters have been
previously fixed using the mass spectra and several low-energy parameters known
from experiment and consistent predictions have been made. As other predictions
of the model, in the present work the isospins $I$= 1/2, 3/2 and $J$=0
projection of $\pi K$ scattering amplitude (as well as phase shifts) are
computed and compared with experiment. In the $I$=1/2 channel, it is shown that
within the uncertainties of the model parameters a good agreement with
experimental data up to an energy of about 1 GeV is obtained, whereas in the
$I$=3/2 channel there is a better agreement with experiment which extends to
about 1.4 GeV. The effect of final state interactions of $\pi K$ in the $I$=1/2
channel is approximated by the K-matrix method and the poles of the unitarized
scattering amplitude are found. It is shown that the model predicts a light and
broad kappa resonance with a mass and decay width of 670-770 MeV and 640-750
MeV consistent with other prior works. Moreover, the scattering lengths in the
$I$=1/2, 3/2 are also computed and shown to qualitatively agree with
experiment. The overall predictions presented here further support previous
findings that the scalar mesons below and above 1 GeV have substantial
underlying mixings and that those below 1 GeV have dominant four-quark
substructures while those above 1 GeV are closer to conventional $P$-wave
quark-antiquark states.
