The considered problem is assessing non-negativity of a function's values when indeterminates are in domains constrained by scalar polynomial inequalities. The tested functions are multi-indeterminates polynomial matrices which are required to be positive semi-definite. New tests based on linear matrix inequalities are provided in a Slack Variables type approach. The results are compared to those obtained via the Sum-Of-Squares approach. They are proved to be equivalent in case of unbounded domains and less conservative if polytopic-type bounds are known.