Summary: In this work, we examine students’ ways of thinking when presented with a novel linear algebra problem. Our intent was to explore how students employ and coordinate three modes of thinking, which we call computational, abstract, and geometric, following similar frameworks proposed by {\it J. Hillel} [in: On the teaching of linear algebra. Dordrecht: Kluwer Academic Publishers. 191‒207 (2000; ME 2016a.00720)] and {\it A. Sierpinska} [in: On the teaching of linear algebra. Dordrecht: Kluwer Academic Publishers. 209‒246 (2000; ME 2016a.00721)]. However, the undergraduate honors linear algebra students in our study used the computational mode of thinking in a surprising variety of productive and reflective ways. This paper examines the solution strategies that the students employed to solve the problem, emphasizing their use of the computational mode of thinking.