  We construct quasi-solvable quantum mechanical matrix models by employing two
different methods, the one is universal enveloping algebra of Lie superalgebra
and the other is N-fold supersymmetry. For the former we examine the q(2) and
osp(2/2) Lie-superalgebraic quasi-solvable matrix operators in the literature,
and then compare them with the corresponding N-fold supersymmetric matrix
systems. In the q(2) case, Lie-superalgebraic construction and the intertwining
relation lead to the identical result. In the osp(2/2) case, however, some
novel features emerge due to the difference in dimension of linear spaces which
consist of the two-component invariant subspace. In both cases, the closure of
N-fold superalgebra imposes stronger constraint on the admissible form of the
systems and the concept of conjugation plays a key role in the formulation.
