determinant
(abbrev. det), n.of a square matrix A is a certain alternating sum of products of entries of A, one from each row and column. Determinants have the property that
det (AB) = det (A) × det (B).
If A is the 2 × 2 matrix
det (A) is equal to ad - bc. The determinant of a matrix of size n can be evaluated as certain linear combinations of determinants of matrices of size n - 1 (for example, its cofactors). For example, the 3 × 3 determinant
Formally, if A = [aij] is an n × n matrix with entries in a commutative unitary ring, then
det (A) = Σσ∈Sn ɛ(σ) a1σ1 a2σ2 ... anσn,
where Sn is the set of permutations of the integers 1 to n, ɛ(σ) is the signature of the permutation σ, and σi is the ith member of the permutation σ. A matrix is invertible if and only if its determinant is non-zero, and determinants may be used in the solution of simultaneous equations, etc. by matrix methods, although Gaussian elimination or related techniques are almost always preferred in actual computation for matrices larger than 3 × 3. Compare permanent.
det (AB) = det (A) × det (B).
If A is the 2 × 2 matrix
det (A) is equal to ad - bc. The determinant of a matrix of size n can be evaluated as certain linear combinations of determinants of matrices of size n - 1 (for example, its cofactors). For example, the 3 × 3 determinant
Formally, if A = [aij] is an n × n matrix with entries in a commutative unitary ring, then
det (A) = Σσ∈Sn ɛ(σ) a1σ1 a2σ2 ... anσn,
where Sn is the set of permutations of the integers 1 to n, ɛ(σ) is the signature of the permutation σ, and σi is the ith member of the permutation σ. A matrix is invertible if and only if its determinant is non-zero, and determinants may be used in the solution of simultaneous equations, etc. by matrix methods, although Gaussian elimination or related techniques are almost always preferred in actual computation for matrices larger than 3 × 3. Compare permanent.