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