There is a matrix, whose components explicitly depend on two variables. For some values of these variables, the determinant vanishes. I need to compute analytically the second derivative of the determinant with respect to these two variables. The inverse matrix does not exist for these values of variables. I found a very nice relationship here Second Derivative of a Determinant However, it is based on the "modified" Jacobi formula for the first derivative, where the adjugate matrix has been replaced by a product of the determinant and the inverse matrix. I cannot apply it, because in my case the inverse matrix does not exist. If we dont'do this replacement, we will need the first derivative of the adjugate matrix. Does a relationship for the derivative of the adjugate matrix exist?
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1Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos May 07 '21 at 07:20
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1Singularity of the matrix is irrelevant to the question. Why do you mention that ? – May 07 '21 at 07:29
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This question is too allusive. You should give explicitly the matrix you are working on. – Jean Marie May 07 '21 at 07:55
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Compute it assuming the matrix is invertible. The final formula can be rewritten without using the inverse matrix. That will be the formula you want. – Deane May 07 '21 at 16:43
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No, the final formula cannot be rewritten without the inverse matrix because the inverse matrix appears in the same monomials twice. Could you please see the link referred in my question? – Diane May 07 '21 at 17:02