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Determinant Meaning: Definition, Examples, and Translations

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determinant

[dษชหˆtษœหrmษชnษ™nt ]

Definition

Context #1 | Noun

mathematics

In mathematics, a determinant is a scalar value that can be computed from the elements of a square matrix. It encapsulates key information about the matrix, including whether the matrix is invertible and the volume scaling factor of the linear transformation described by the matrix. The determinant is denoted as det(A) for a matrix A and can be calculated using various methods, including cofactor expansion or row reduction. Determinants are particularly significant in solving systems of linear equations and in linear algebra applications.

Synonyms

determinative, factor.

Examples of usage

  • To determine if the matrix is invertible, calculate its determinant.
  • The area of the parallelogram can be found using the determinant of vectors.
  • The determinant of the identity matrix is always one.

Translations

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Interesting Facts

Mathematics

  • Determinants are used in linear algebra to understand properties of matrices, such as whether a system of equations has a unique solution.
  • The determinant gives information about the volume of a shape in higher-dimensional space; if it's zero, the shape flattens out.
  • This concept was first rigorously described by mathematician Gottfried Wilhelm Leibniz in the 17th century.

Philosophy

  • In philosophical discussions, determinism refers to the idea that all events are determined by preceding causes, paralleling the mathematical sense of a determinant providing outcomes.
  • Some philosophers debate whether free will exists if every action can be predicted by prior states, similar to how a determinant predicts solutions.

Computer Science

  • In computer science, determinants show up in algorithms involving matrices, especially within graphics and machine learning.
  • Efficient computation of determinants is crucial for solving computational problems, such as those found in optimization.

Education

  • Teaching about determinants can help students understand concepts of area, linear transformations, and matrix operations in real-world scenarios.
  • Determinants are often a challenging topic for students but are pivotal for advancing in mathematics education.

Pop Culture

  • In popular shows and films, themes involving determinism often question the destiny versus free will, echoing the way determinants provide predictable outcomes.
  • Shows like 'Dark' explore timelines and choices, reflecting the philosophical implications associated with the concept of determinism.

Origin of 'determinant'

The word 'determinant' first appeared in English in the 19th century, deriving from the Latin root 'determinare', meaning 'to limit' or 'to define'. Its mathematical usage has roots in linear algebra, where the concept was developed to provide a way to understand the properties of square matrices. The term reflects its function in mathematics as a defining factor or a limit to the properties of linear transformations. Over time, 'determinant' has become a standard term in mathematics, particularly in discussions involving matrices and linear equations, emphasizing its fundamental role in the field.

Word Frequency Rank

Ranked #10,905, this word falls into high-advanced vocabulary. It appears less frequently but is valuable for expressing precise meanings in specific contexts.