Nonlinear non-abelian Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg–de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties Carillo and Schiebold (J Math Phys 50:073510, 2009) enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations allow to construct explicit solution formulae Carillo and Schiebold (J Math Phys 52:053507, 2011). The latter are adapted to obtain solutions admitted by the 2 × 2 and 3 × 3 matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.