invertible

/in'və:təbl/

Thân thiện

The mathematician writes an invertible matrix on the chalkboard.

Definition

1. Adjective:
   • Having an additive or multiplicative inverse: In mathematics, an element is described as "invertible" if there exists another element that, when combined with it using a specific operation (like addition or multiplication), yields the identity element for that operation. For example, in multiplication, the number 2 is invertible because 2 × (1/2) = 1, where 1 is the multiplicative identity.

Usage Examples

• Adjective:
  • In the set of real numbers, every non-zero number is invertible under multiplication.
  • The matrix is invertible because its determinant is not zero, meaning it has a multiplicative inverse.
  • For a function to have an inverse function, it must be invertible; that is, it must be one-to-one.

Advanced Usage

• Invertible Function: A function is called invertible if it is bijective (both one-to-one and onto), allowing for the definition of an inverse function that reverses its mapping.

  • The linear function f(x) = 3x + 1 is invertible, and its inverse is f⁻¹(x) = (x - 1)/3.

• Invertible Element in a Ring: In abstract algebra, an element a in a ring R is invertible if there exists an element b in R such that a·b = b·a = 1, where 1 is the multiplicative identity in R.

  • In the ring of integers modulo 5, the element 3 is invertible because 3 × 2 ≡ 1 (mod 5).

Variants and Related Words

• Invert (verb): To turn something upside down or reverse its position, order, or relationship.

  • You can invert the image in the photo editor.

• Inverse (noun/adjective): Something that is the direct opposite; in mathematics, the element that produces the identity when combined with a given element.

  • Division is the inverse operation of multiplication.

• Invertibility (noun): The property or condition of being invertible.

  • The invertibility of the matrix is crucial for solving the system of equations.

Synonyms

• Reversible: Capable of being reversed or inverted.
• Nonsingular: (Specifically for matrices) Having a non-zero determinant; invertible.

Antonyms

• Non-invertible: Lacking an inverse.
• Singular: (Specifically for matrices) Having a determinant of zero; not invertible.

Học thuật

Adjective

1. having an additive or multiplicative inverse