Single-step GBLUP (ssGBLUP) is a genomic prediction method combining phenotypes, pedigree information, and genotypes in a system of equations. This method needs the inverse of unified relationships (H⁻¹) as a function of numerator relationship matrix for all animals in pedigre (A⁻¹), genomic relationship matrix (G⁻¹), and numerator relationship matrix for genotyped animals (A₂₂⁻¹). Although these matrices should be compatible in scale, it is rarely true in dairy-cattle data because the pedigree is long but partially missing and the genotypes are mainly available for the last few generations. Unknown parent-groups (UPG) and metafounders (MF) can be used to fill the missing relationships in the relationship matrices and to make them compatible in scale. The objectives of this study were to show a reasonable relationship matrix with UPG or MF in ssGBLUP, to implement this matrix in genetic prediction software, and to apply this technique to production traits in US Holstein. We have derived H⁻¹ in which UPG are considered only for pedigree relationships. We also derived an alternative H⁻¹ from a joint density function. This alternative form is equivalent to one derived from the MF theory. Although UPG and MF are equivalent under some assumptions, MF seems to be more flexible to keep the compatibility among the relationship matrices. The alternative H⁻¹ showed that the computation of A₂₂⁻¹ with MF is greatly simplified with sparse matrix techniques. The newly-derived H⁻¹ was implemented in the BLUPF90 programs. The genotypes, pedigrees, and phenotypes of 305-d milk, fat, and protein yield were provided by the Council of Dairy Cattle Breeding (CDCB). The data included more than 72 million phenotypes for each trait, 80 million pedigree animals, and 2.3 million genotyped animals. The validation predictability and inflation of genomic predictions with UPG or MF will be presented.