Determinant

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The determinant, a function which associates a number to every square matrix, is used to calculate volumes in vector calculus, to characterize invertible matrices, and to explicitly describe the solution to a system of linear equations with Cramer's rule. The determinant of the square matrix A is denoted by det(A) or |A|. One may also view the determinant as a function that associates a number to a sequence of n vectors from Rn (see real numbers).

= Computing determinants

If A is a 1-by-1 matrix, then det(A) = A1,1. If A is a 2-by-2 matrix, then det(A) = A1,1 · A2,2 - A2,1 · A1,2. For a 3-by-3 matrix A, the formula is more complicated:

det(A) = A1,1·A2,2·A3,3 + A1,3·A3,2·A2,1 + A1,2·A2,3·A3,1
- A3,1·A2,2·A1,3 - A1,1·A2,3·A3,2 - A1,2·A2,1·A3,3

In general, determinants can be computed with the Gauss algorithm using the following rules:

  • If A is a triangular matrix, i.e. Ai,j = 0 whenver i > j, then det(A) = A1,1·A2,2·...·An,n
  • If B results from A by exchanging two rows or columns, then det(B) = - det(A)
  • If B results from A by multiplying one row or column with the number c, then det(B) = c · det(A)
  • If B results from A by adding a multiple of one row or column to another row or column, then det(B) = det(A).

Explicitely, use the above rules to construct a triangular matrix, then compute the result from the first rule.

Compatibility with matrix multiplication

The determinant function is compatible with matrix multiplication in the following sense: if A and B are square matrices of the same size, then det(AB) = det(A) · det(B). Furthermore, A is invertible if and only if det(A) ≠ 0; if this is the case, then det(A-1) = det(A)-1.

Interpretation of the determinant for real vectors

The sign of the determinant of real vectors has a special significance because it serves to define the notion of orientation of coordinate systems. If three vectors in R3 are given, then they may be oriented similarly to the first three fingers of the right hand, in which case their determinant will be positive, or similarly to the first three fingers of the left hand, in which case their determinant will be negative. A similar statement holds true for higher dimensions.

The absolute value of the determinant of real vectors is important in volume computations because it is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map f : Rn -> Rn is represented by the matrix A, and S is any subset of Rn, then the volume of f(S) is given by |det(A)| * volume(S). More generally, if the linear map f : Rn -> Rm is represented by the m-by-n matrix A, and S is any subset of Rn, then the n-dimensional volume of f(S) is given by sqrt(det(Atr * A)) * volume(S), where Atr denotes the transpose of A.

Generalization

If f : V -> V is a linear transformation (also called "endomorphism") of a finite dimensional vector space, we may define its determinant det(f) by first picking a basis of V, then representing f as a matrix with respect to that basis, and then computing the determinant of that matrix. This determinant will only depend on f and not depend on the basis chosen.

It makes sense to define the determinant for matrices whose entries come from any commutative ring, and in particular from any field. The computation rules and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if det(A) is invertible in the ground ring.


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