Let \(R\) be a commutative ring, \(J\) its Jacobson radical and \(E(R/J)\) the injective envelope. \(A^*:=\text{Hom}_ R(A,E(R/J))\) is called the \(G\)- Matlis dual of the \(R\)-module \(A\), and \(A\) is \(G\)-Matlis reflexive if the canonical map \(\to A^{**}\) is an isomorphism. The authors study relations between the homological dimension of \(A\) and \(A^*\). If \(R\) is coherent resp. noetherian complete semilocal and \(A\) \(G\)-Matlis reflexive resp. artinian, the injective dimension of \(A\) is equal to the weak homological resp. projective dimension of \(A^*\).