detBT

detBT denotes the determinant of the product of two square matrices B and T of the same size. It is a scalar quantity associated with the linear transformation obtained by composing the maps represented by B and T.

In standard linear algebra, the determinant is multiplicative: det(BT) = det(B) det(T). This holds for all n x n matrices over a field such as the real or complex numbers. Consequently, det(BT) = 0 if either det(B) = 0 or det(T) = 0. If both B and T are invertible, then det(BT) ≠ 0, and det((BT)^{-1}) = 1/(det(B) det(T)). The equality det(BA) = det(AB) implies the same for the reverse product.

The determinant of a product depends only on the determinants of the factors, not on their individual entries. This property is often used to simplify computations, since one can obtain det(BT) directly from det(B) and det(T) without forming the product matrix. It also explains how volume scaling behaves under the composition of linear transformations: det(BT) gives the overall volume scaling factor of applying T followed by B.

Notes and related concepts: If you know det(B) and det(T), you can compute det(BT) without multiplying the matrices. Conversely, forming BT and computing its determinant yields det(BT) even when det(B) or det(T) are difficult to determine separately. Related ideas include the determinant of a product, eigenvalues of product matrices, and properties such as det(B^T) = det(B).