Definition

1. Adjective:
   • Not admitting an additive or multiplicative inverse: In mathematics, a matrix or an element in a ring is described as "non-invertible" if there does not exist another matrix or element that, when combined with it through the relevant operation (addition or multiplication), yields the identity element (zero for addition, one for multiplication).

Usage

• The term "non-invertible" is primarily used in formal mathematical contexts, specifically in linear algebra and abstract algebra.
• It describes a fundamental property of a mathematical object, indicating it cannot be "undone" or reversed by a corresponding inverse operation within the given system.

Examples

• Adjective:
  • The matrix had a determinant of zero, making it non-invertible.
  • In modular arithmetic, the element 2 is non-invertible modulo 4 because there is no integer x such that 2x ≡ 1 (mod 4).
  • A function is non-invertible if it is not one-to-one.

Advanced Usage

• "Non-invertible transformation": In linear algebra, a linear transformation that cannot be reversed because it maps a space to one of lower dimension (its kernel is non-trivial).
  • Projection onto a line is a classic example of a non-invertible transformation.
• "Non-invertible element": In ring theory, an element that does not have a multiplicative inverse within the ring.
  • The integer 0 is a non-invertible element in the ring of integers.

Variants and Related Words

• Singular (adj): A common synonym for a non-invertible matrix. ()
• Degenerate (adj): Sometimes used interchangeably with "singular" or "non-invertible" for matrices.
• Invertible (adj): The direct antonym, meaning an inverse does exist.

Synonyms

• Singular
• Degenerate (for matrices)
• Non-singular (This is the antonym; it is listed here for contrast as it is the more common term in many contexts.)

Related Phrases

• "To be non-invertible": The standard phrase describing the state.
  • A square matrix is non-invertible if and only if its determinant is zero.

Adjective

1. not admitting an additive or multiplicative inverse