Let A∈F n×nA\in\mathbb{F}^{\,n\times n} be a square matrix. If the ithi^{\mathrm{th}} row and jthj^{\mathrm{th}} column is removed from AA the resulting matrix Bij∈F (n−1)×(n−1)B_{ij} \in \mathbb{F}^{\,(n-1)\times (n-1)} is called the (i,j)(i,j)-sub-matrix of A.A. The (i,j)(i,j)-minor of AA is the determinant Mij∈FM_{ij}\in\mathbb{F} of the (i,j)(i,j)-sub-matrix Bij.B_{ij}. The (i,j)(i,j)-co-factor of A,A, Cij∈F,C_{ij}\in\mathbb{F}, is the (i,j)(i,j)-minor MijM_{ij} scaled by (−1)i+j.(-1)^{i+j}.

Mij=∣Bij∣M_{ij} = \lvert B_{ij}\rvert Cij=(−1)i+jMijC_{ij} = (-1)^{i+j}M_{ij}

The determinant, det(A)\mathrm{det}(A) or ∣A∣,\lvert A\rvert, of AA is defined recursively. For n=1,n=1, ∣A∣=a11.\lvert A \rvert = a_{11}. For n≥2n \geq 2 it is defined with the Laplace expansion of AA along any fixed row i∈{1,…,n}:i \in \{1,\ldots,n\}:

∣A∣=∑j=1naijCij\lvert A \rvert = \sum_{j=1}^{n} a_{ij}C_{ij}

Every choice of row ii gives the same value.

For the simple case n=2,n=2, ∣A∣=a11a22−a12a21.\lvert A \rvert = a_{11}a_{22}-a_{12}a_{21}.

The determinant of a real, square matrix A∈R n×nA\in\mathbb{R}^{\,n\times n} is the signed volume of the parallelepiped formed by taking each row of AA as a vector in Rn.\mathbb{R}^n. The volume itself is ∣det(A)∣\lvert \mathrm{det}(A) \rvert and the sign records whether the rows have the same orientation as the standard basis.

The determinant of square matrices A,B∈F n×nA,B\in\mathbb{F}^{\,n\times n} satisfies the following properties:

  • Determinant of transpose: ∣A∣ =∣A⊺∣\lvert A \rvert\,= \lvert A^\intercal \rvert
  • Determinant of scalar multiplication: ∣λA∣ =λn∣A∣  ∀ λ∈F\lvert \lambda A \rvert\,= \lambda^n\lvert A \rvert\,\,\forall\,\lambda\in\mathbb{F}
  • Determinant of matrix multiplication: ∣AB∣ =∣A∣∣B∣\lvert AB \rvert\,= \lvert A \rvert \lvert B \rvert
  • Determinant of matrix inverse: ∣A−1∣ =1∣A∣\lvert A^{-1} \rvert\,=\frac{1}{\lvert A \rvert} if AA is invertible
  • Invertibility: ∣A∣ ≠0⇔A\lvert A \rvert \,\neq 0 \Leftrightarrow A is invertible
  • Full rank: ∣A∣ ≠0⇔A\lvert A \rvert \,\neq 0 \Leftrightarrow A has rank nn

Let A∈F n×nA\in\mathbb{F}^{\,n\times n} be an invertible square matrix, so that ∣A∣≠0,\lvert A\rvert \neq 0, and let C∈F n×nC\in\mathbb{F}^{\,n\times n} be its co-factor matrix whose (i,j)th(i,j)^{\mathrm{th}} element is Cij.C_{ij}. The matrix inverse A−1A^{-1} can be calculated from the determinant and co-factors of A:A:

A−1=C⊺∣A∣A^{-1}=\frac{C^\intercal}{\lvert A\rvert}

where the (i,j)th(i,j)^{\mathrm{th}} element of A−1A^{-1} is Cji∣A∣.\frac{C_{ji}}{\lvert A\rvert}.