Residuals of Boolean valued relations were introduced by G. Birkhoff in 1948 as the best possible approximation to an inverse in the monoid of relations. We have generalized these concepts by introducing residuals of fuzzy relations. We show that residuals of fuzzy relations play a crucial role in computatuion of the greatest solutions to some systems of fuzzy relation inequalities and equations with significant applications in various fields, e.g., in automata theory, network analysis, etc. In general there is no residuation for matrices over a semiring, but we show that for matrices over an additively idempotent semiring there is a kind of relative residuation which allows to define and study Boolean residuals of matrices.