Non-singular matrices have a key role to play when determining the rank of a matrix. The rank of a matrix is useful when deducing the nature of solutions to a system of linear equations. In this paper I report on a qualitative case study which explored possible mathematical gaps that created barrier/s when engineering students (n=101) worked with the rank of matrices to determine the nature of the solutions of a system of linear equations. These engineering students were provided with two tasks and their written responses were used to detect possible barriers in their understanding of the rank concept. The study was carried out at a University of Technology in South Africa. The data, which was collected from written responses of students to two tasks, were analysed using APOS theory. Interviews followed these analyses to verify the APOS levels they were placed at. Findings emanating from the data analysis indicated that we: 1) can correctly place the APO level of conception of an individual if the individual provides a completely mathematically correct solution, 2) cannot be certain at which mental construction an individual is performing at when partially mathematically correct responses are provided and 3) individuals providing completely incorrect solutions require further exploration.