detATdetA

detATdetA denotes the product det(A^T) det(A) for a square matrix A. Since det(A^T) = det(A) for every square A, detATdetA equals det(A)^2. Equivalently, detATdetA = det(A^T A), because det(XY) = det(X) det(Y). This identity holds for all n×n matrices A (with det(A) defined).

For square A, detATdetA is zero exactly when A is singular; if A has real entries, det(A) is real, so detATdetA = det(A)^2 is a nonnegative real number. More generally, det(A^T A) equals the product of the squares of A’s singular values, and in particular det(A^T A) = det(A)^2 when A is square and invertible.

Geometrically, det(A) measures the oriented volume (or area in two dimensions) scaling of the linear transformation represented by A. Therefore det(A)^2, i.e., detATdetA, corresponds to the squared volume scaling. The matrix A^T A is the Gram matrix of the column vectors of A, and det(A^T A) is the Gram determinant, equal to the squared volume of the parallelepiped spanned by those columns.

Example: Let A = [[2,0],[0,3]]. Then det(A) = 6, det(A^T) det(A) = 36, and det(A^T A) = 36. See also determinants, transpose, Gram matrix, singular values.