Jacobi's formula - Wikipedia
en.wikipedia.org › wiki › Jacobi&In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. ∂ det ( A ) ∂ A i j = adj ( A ) j i . {\displaystyle {\partial \det (A) \over \partial A_ {ij}}=\operatorname {adj} (A)_ {ji}.}
Jacobi’s formula for the derivative of a determinant
gotohaggstrom.com › Jacobis formula for theTo calculate the derivative of the determinant we use (13) with: f(A) = f(a 11;a 12;a 21;a 22) = a 11a 22 a 21a 12 (18) Thus: @f @a 11 =a 22 @f @a 12 = a 21 @f @a 21 = a 12 @f @a 22 =a 11 (19) Also: da 11 dt =2 da 12 dt =1 da 21 dt = 1 da 22 dt =3 (20) So formula (13) gives: df dt = X2 i=1 X2 j=1 @f @a ij da ij dt = 3t 2 + t 1 + ( t) ( 1) + 2t 3 = 14t (21) From (14) d dt A = 14t. Thus con rming (1): jAj jAj = 14t 7t2 =
Jacobian matrix and determinant - Wikipedia
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinantIn vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.