Derivative of the inverse of a matrix

In a scientific paper, I’ve seen the following


where K is a n×n matrix that depends on p. In my calculations I would have done the following


Is my calculation wrong?

Note: I think K is symmetric.


The major trouble in matrix calculus is that the things are no longer commuting, but one tends to use formulae from the scalar function calculus like (x(t)1)=x(t)2x(t) replacing x with the matrix K. One has to be more careful here and pay attention to the order. The easiest way to get the derivative of the inverse is to derivate the identity I=KK1 respecting the order
Solving this equation with respect to (K1) (again paying attention to the order (!)) will give

Source : Link , Question Author : Sara , Answer Author : A.Γ.

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