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

Construction of Quasi-solvable Quantum Mechanical Matrix Models: Lie Superalgebra v.s. N-fold Supersymmetry