2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124044New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer rule for right and left quaternionic systems of linear equations have been obtained.34 pages, submitted to Fundamentalnya i prikladnya matematika (Fundamental and Applied Mathematics)Rings and AlgebrasRepresentation Theory15A06, 15A15, 15A33text