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Second Derivative of Determinant of Matrix?

by brydustin
Tags: derivative, determinant, matrix
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Dec31-11, 03:51 PM
P: 209
Hi all....

I've read on wikipedia (facepalm) that the first derivative of a determinant is

del(det(A))/del(A_ij) = det(A)*(inv(A))_j,i

If we go to find the second derivative (applying power rule), we get:

del^2(A) / (del(A)_pq) (del (A)_ij) = {del(det(A))/del(A_pq)}*(inv(A))_j,i + det(A)*{del(inv(A)_j,i) / del(A_pq)}

I have no clue how to calculate the derivative of the inverse of a matrix with respect to changing the values in the original matrix:
I.E. del(inv(A)_j,i) / del(A_pq)

Also.... would be nice if someone could prove the first statement for the first derivative of the determinant.

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