Little is known about the variety of ways students conceptualize matrix multiplication, yet this is a fundamental part of most introductory linear algebra courses. My dissertation follows a three-paper format, with the three papers exploring conceptualizations of matrix multiplication from a variety of viewpoints. In these papers, I explore (1) the historical roots of matrix multiplication as they arose through theory about systems of linear equations, (2) the variety of ways students think about matrix multiplication, and (3) the informal and intuitive ways students model real-world situations that can be represented using matrix multiplication. The first paper is a theoretical piece in which I discuss historic contexts that gave rise to considerations of linear systems of equations and their solutions, the ways in which available tools and representations shaped the development of mathematical ideas in those contexts, and the pedagogical implications of this analysis. In the second paper I develop a framework for understanding student thinking about matrix multiplication. Three categories of student conceptualizations emerged from my analysis: "matrix acting on a vector" views, "vector acting on a matrix" views, and "systems of linear equations" views. The framework coordinates these three categories with three contexts of matrix multiplication, where the curtailment of notation and the presence or absence of an equals sign varied across contexts. Data for this paper was taken from a set of semi-structured clinical interviews conducted with students enrolled in an undergraduate inquiry-oriented introductory linear algebra class. The data was analyzed using a combination of grounded theory and the constant comparative method. In the final paper I illustrate how quantitative reasoning can be used as a lens to analyze students' modeling of a real-world problem situation that can be approached using the tools of linear algebra. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.