using LinearAlgebra
A = rand(3,3)

A2 = A^2

dA = [0.001 0.002 0.003 ; 0.004 0.005 0.006 ; 0.007 0.008 0.009]

(A + dA)^2 - A2

A*dA + dA*A

2*A*dA

2*dA*A

AI = inv(A)

inv(A+dA)-inv(A)

-AI*dA*AI