Let y be observation vector in the usual linear model with expectation Aβ and covariance matrix known up to a multiplicative scalar, possibly singular. A linear statistic aTy is called invariant estimator for a parametric function ϕ=cTβ if its MSE depends on β only through ϕ. It is shown that aTy is admissible invariant for ϕ, if and only if, it is a BLUE of ϕ, in the case when ϕ is estimable with zero variance, and it is of the form kϕˆ, where k∈⟨0,1⟩ and ϕˆ is an arbitrary BLUE, otherwise. This result is used in the one- and two-way ANOVA models. Our paper is self-contained and accessible, also for non-specialists.