##  [Determinant](/determinant) 

  ##  [Determinant](https://natural.quantumdictionary.io/determinant-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A scalar-valued function det: M_n(F) → F on the n×n square matrices over a field or commutative ring F characterized by multilinearity in rows (or columns), alternation (sign change under row swap) and normalization det(I)=1; it measures oriented volume scaling and decides invertibility.

 

 

 

 

 





 

 



 ##  [Determinant](https://mathlogic.quantumdictionary.io/determinant-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A scalar-valued function assigned to a square linear operator or matrix that records oriented volume scaling and indicates invertibility; algebraically it is an alternating multilinear form of the columns (or rows).

 

 

 

 

 





 

 



 ##  [Determinant](https://puremath.quantumdictionary.io/determinant-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A scalar-valued polynomial invariant assigned to a square matrix or endomorphism that encodes whether the linear map is invertible and how oriented volumes are scaled by the map.

 

 

 

 

 





 

 



 ##  [Determinant](https://algebra.quantumdictionary.io/determinant-3) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A scalar invariant associated to a square matrix or endomorphism that encodes volume scaling factor, orientation sign, and invertibility: a matrix is invertible iff its determinant is nonzero.