Determinant (mathematics)
The determinant is a numerical value associated with square matrices that plays a significant role in various areas of mathematics and its applications. It is crucial for solving systems of linear equations, where a nonzero determinant indicates that a unique solution exists. Geometrically, the absolute value of a determinant corresponds to the area or volume of geometric shapes formed by the matrix, such as parallelograms and parallelepipeds. To compute a determinant, the matrix must have an equal number of rows and columns, and different formulas apply depending on the size of the matrix, such as simple multiplication for 2 × 2 matrices and more complex expressions for larger matrices.
Historically, the concept of determinants has roots in ancient mathematical texts, with significant contributions from figures like Girolamo Cardan and Gabriel Cramer, who explored methods for solving linear systems. The formal usage of determinants as we understand them today was advanced by mathematicians such as Gottfried Wilhelm von Leibniz and Seki Takakazu in the 17th century. While historical computing methods for determinants have largely been supplanted by modern software, their relevance persists in fields like multivariable calculus and quantum chemistry. Overall, the determinant remains a fundamental tool in both theoretical and applied mathematics, facilitating the analysis of linear transformations and real-world phenomena.
On this Page
Subject Terms
Determinant (mathematics)
The determinant is a number associated to a square matrix, which has wide applicability both inside and outside of mathematics. The determinant is important because it is useful in helping to understand systems of linear equations. A nonzero determinant has a particularly special role because these systems have exactly one solution, and the determinant appears algebraically as a factor in the solution. A determinant is also important from a physical or geometric viewpoint because the absolute value of the determinant represents the area or volume of related parallelograms or parallelepipeds. In addition, the determinant provides geometric information about the linear transformation that corresponds to a matrix.
A matrix is an array, which has numbers or objects organized in rows and columns. To compute a determinant, a matrix must have the same number of rows as columns. For a 2 × 2 matrix, the determinant is the multiplication of the entries on the main diagonal minus the multiplication of those on the off diagonal. For example, the determinant of
is 1 × 4 – 2 × 3 = −2. In general, the determinant is a numerical expression involving additions and subtractions of certain products of entries, like for the 3 × 3 matrix
, whose determinant is aei + bfg + cdh – gec – hfa – idb.
Overview
Mathematics historians point to a number of precursors of determinants. The ancient Chinese handbook The Nine Chapters of the Mathematical Art focused on mathematical methods useful in real life. Some solutions to linear systems in this work use a determinant-like multiplication. In the sixteenth century, Italian physician and mathematician Girolamo Cardan investigated a version of what we now call Cramer’s rule for 2 × 2 matrices, which solves systems using determinants. It is named for Swiss mathematician Gabriel Cramer, who worked on n × n systems.
It is German mathematician Gottfried Wilhelm von Leibniz and Japanese mathematician Seki Takakazu who are generally credited with discovering determinants. They were working independently in the seventeenth century. Seki investigated solutions to equations as well as what we now call determinants of 2 × 2 matrices all the way up to 5 × 5 matrices. Leibniz investigated many results related to determinants, but these were mostly in unpublished works. For example, he studied Cramer’s rule as well as precursors to Laplace’s expansion of the determinant, also known as the cofactor expansion, which was eventually named for French mathematician and astronomer Pierre-Simon Laplace.
French mathematician Augustin-Louis Cauchy first used determinants in the present-day sense in 1812. By the beginning of the twentieth century, Scottish mathematician Sir Thomas Muir had published five volumes on the history of determinants. Methods of computing determinants by hand were once a larger focus of linear algebra, before the advent of computer algebra software systems that can calculate determinants. In fact, some mathematicians have cautioned against an overreliance on determinants. However, determinants remain important in other ways, for example, the change of variable formula for integrals in multivariable calculus and Slater determinants used in quantum chemistry, named for American physicist John C. Slater. Applications can be found in many fields where real-life scenarios are modeled by linear systems.
Bibliography
Anton, Howard, and Chris Rorres. Elementary Linear Algebra: Applications Version. 11th ed. New York: Wiley, 2013.
Axler, Sheldon. "Down with Determinants!" American Mathematical Monthly 102.2 (1995): 139-154.
Benjamin, Arthur T., and Naiomi T. Cameron. "Counting on Determinants." American Mathematical Monthly 112.6 (2005): 481-492.
Bremner, Murray R. "Matrices." Encyclopedia of Mathematics and Society. Eds. Sarah J. Greenwald and Jill E. Thomley. Pasadena, CA: Salem Press, 2011.
Bressoud, David M. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Washington, DC: Mathematical Assoc. of America, 1999.
Cheney, Ward, and David R. Kincaid. Linear Algebra: Theory and Applications. 2nd ed. Sudbury, MA: Jones, 2012.
Debnath, Lokenath. "A Brief Historical Introduction to Determinants with Applications." International Journal of Mathematical Education in Science and Technology 44.3 (2013): 388-407.
Hart, Roger. The Chinese Roots of Linear Algebra. Baltimore, MD: The Johns Hopkins UP, 2011.