  We give two algorithms to compute linear determinantal representations of
smooth plane curves of any degree over any field. As particular examples, we
explicitly give representatives of all equivalence classes of linear
determinantal representations of two special quartics over the field
$\mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic.
This paper is a summary of third author's talk at the JSIAM JANT workshop on
algorithmic number theory in March 2018. Details will appear elsewhere.
