Linear Algebra/Determinant

The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these properties:

  • It is linear on the rows of the matrix.
  • If the matrix has two equal rows its determinant is zero.
  • The determinant of the identity matrix is 1.

It is possible to prove that , making the definition of the determinant on the rows equal to the one on the columns.


  • The determinant is zero if and only if the rows are linearly dependent.
  • Changing two rows changes the sign of the determinant:

  • The determinant is a multiplicative map in the sense that
  for all n-by-n matrices   and  .

This is generalized by the Cauchy-Binet formula to products of non-square matrices.

  • It is easy to see that   and thus
  for all  -by-  matrices   and all scalars  .
  • A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have

Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.

A matrix and its transpose have the same determinant:


The determinants of a complex matrix and of its conjugate transpose are conjugate:








Using Laplace's formula for the determinant

Binet's theorem