Матрицы и детерминанты: структура, преобразование и разрешимость

Matrices and Determinants presents matrix theory as a structured mathematical language for transformation, systems of equations, and solvability, rather than as a collection of computational rules. Beginning with the basic definitions and algebra of matrices, the book develops matrix operations, types of matrices, and elementary row operations in a logical progression. Determinants are introduced as structural objects that capture invertibility, area, volume, and solvability of linear systems, with careful attention to their properties and interpretation. Emphasis is placed on understanding how matrices represent linear transformations, how determinants govern system behavior, and how these ideas connect to real-world applications and higher mathematics. Methods for solving systems of linear equations, including matrix inversion and row-reduction techniques, are presented in a clear and method-driven manner. Throughout the book, worked examples, graded exercises, and discussions of common errors help readers develop both computational accuracy and conceptual clarity. The treatment builds a strong foundation for further study in linear algebra, vector spaces, calculus, and applied mathematics. This book is written for high achievers in IB, AP, and A-Level mathematics, advanced high-school and early undergraduate students, and learners aiming for the world’s top universities, where a solid understanding of linear structures and mathematical reasoning is essential. This volume forms part of the Math By Rishabh Series, which emphasizes depth, structure, and rigorous problem solving across core areas of mathematics.