Square Matrices of Order 2

1. Front Matter

   Pages i-xxii

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2. Matrices of order 2

   • Vasile Pop, Ovidiu Furdui

   Pages 1-62

3. The Cayley–Hamilton Theorem

   • Vasile Pop, Ovidiu Furdui

   Pages 63-106

4. Applications of Cayley–Hamilton Theorem

   • Vasile Pop, Ovidiu Furdui

   Pages 107-182

5. Functions of matrices. Matrix calculus

   • Vasile Pop, Ovidiu Furdui

   Pages 183-280

6. Applications of matrices to plane geometry

   • Vasile Pop, Ovidiu Furdui

   Pages 281-313

7. Conics

   • Vasile Pop, Ovidiu Furdui

   Pages 315-345

8. Back Matter

   Pages 347-370

   PDF

This unique and innovative book presents an exciting and complete detail of all the important topics related to the theory of square matrices of order 2. The readers exploring every detailed aspect of matrix theory are gently led toward understanding advanced topics. They will follow every notion of matrix theory with ease, accumulating a thorough understanding of algebraic and geometric aspects of matrices of order 2. The prime jewel of this book is its offering of an unusual collection of problems, theoretically motivated, most of which are new, original, and seeing the light of publication for the first time in the literature. Nearly all of the exercises are presented with detailed solutions and vary in difficulty from easy to more advanced. Many problems are particularly challenging. These, and not only these, invite the reader to unleash their creativity and research capabilities and to discover their own methods of attacking a problem.

Matrices have a vast practical importance to mathematics, science, and engineering; therefore the readership of this book is intended to be broad: high school students wishing to learn the fundamentals of matrix theory, first year students who like to participate in mathematical competitions, graduate students who want to learn more about an application of a certain technique, doctoral students who are preparing for their prelim exams in linear algebra, and linear algebra instructors. Chapters 1–3 complement a standard linear algebra course. Pure and applied mathematicians who use matrix theory for their applications will find this book useful as a refresher. In fact, anyone who is willing to explore the methodologies discussed in this book and work through a collection of problems involving matrices of order 2 will be enriched.

• special matrices
• problems matrix theory
• elementary transformations
• Cayley-Hamilton theorem
• Jordan canonical theorem
• powers of matrices
• binomial matrix
• Pell's Diophantine equation
• matrix Gamma function
• affine applications
• homothetic transformations
• degenerate conics
• nondegenerate conics
• series mirabilis
• matrix calculus
• functions of matrices
• conics
• homographic sequences
• characteristic polynomial
• matrix Riemann zeta function

“The book is written in a clear and precise style. The book contains a vast collection of problems varying from easy to difficult levels and their detailed solutions, which help the reader to learn some basic and advanced linear algebra. … The book will be interesting for people who like matrix theory and its challenging problems; in particular, for students who are preparing themselves for mathematical competitions.” (Mohammad Sal Moslehian, Mathematical Reviews, September, 2017)



“This is a book that contains almost everything you would like to know about 2 x 2 matrices. … the book is addressed mainly to advanced high school students, undergraduate and graduate students and professionals according to their needs for results about 2 x 2 matrices.” (Andreas Arvanitoyeorgos, zbMATH 1367.15001, 2017) 

• Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

  Vasile Pop, Ovidiu Furdui

Vasile Pop is an associate professor of mathematics at the Technical University of Cluj-Napoca, Romania. He teaches linear algebra and functional equation classes and coaches students for SEEMOUS as well as high school students for the International Mathematical Olympiad.

Ovidiu Furdui is an associate professor of mathematics at the Technical University of Cluj-Napoca, Romania. He has published more than 300 problems in journals, with a problem column, from all over the world and he is the author of the Springer text Limits, Series and Fractional Part Integrals.

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