Statistics, game theory, economic models, electronic circuits, image processing, computer graphics, quantum mechanics, cryptology, control systems, deep learning, neural networks: all, and many more, make use of matrices and linear algebra. But what are matrices? How to compute with them? How to use them? Assuming almost no prior knowledge in mathematics beyond what is taught at elementary school, students (of computer science, engineering, life science, economics, finance, mathematics, etc.) are introduced stepwise to matrices via such concrete questions. The learning-by-doing approach, the focus on computation, and the emphasis on practice taken in this self-contained book aim not only at teaching students how to address these issues efficiently. Foremost, their "raison d'être" is to allow students to understand what they do, and why.The author presents in a very clear way some of the most relevant algorithms in linear algebra. While literature and pictural art punctuate its chapters, this elegantly written book shows that order matters in the universe of matrices.