In [2]:

using LinearAlgebra
A = rand(3,3)

Out[2]:

3×3 Array{Float64,2}:
 0.262707  0.722601  0.227894
 0.194994  0.545517  0.595722
 0.465314  0.686887  0.644692

In [3]:

A2 = A^2

Out[3]:

3×3 Array{Float64,2}:
 0.31596   0.740561  0.63726 
 0.434797  0.847685  0.753471
 0.556164  1.15377   0.930863

In [4]:

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

Out[4]:

3×3 Array{Float64,2}:
 0.001  0.002  0.003
 0.004  0.005  0.006
 0.007  0.008  0.009

In [5]:

(A + dA)^2 - A2

Out[5]:

3×3 Array{Float64,2}:
 0.00682701  0.00987187  0.0105702
 0.0114308   0.0177037   0.0170739
 0.0154144   0.0252529   0.0236327

In [6]:

A*dA + dA*A

Out[6]:

3×3 Array{Float64,2}:
 0.00679701  0.00983587  0.0105282
 0.0113648   0.0176227   0.0169779
 0.0153124   0.0251269   0.0234827

In [7]:

2*A*dA

Out[7]:

3×3 Array{Float64,2}:
 0.00949674  0.0119231  0.0143495
 0.0130942   0.0157667  0.0184392
 0.0154514   0.0190452  0.022639

In [8]:

2*dA*A

Out[8]:

3×3 Array{Float64,2}:
 0.00409727  0.00774859  0.00670683
 0.00963536  0.0194786   0.0155167 
 0.0151735   0.0312086   0.0243265

In [9]:

AI = inv(A)

Out[9]:

3×3 Array{Float64,2}:
 -0.85783  -4.61434    4.56708  
  2.25985   0.944638  -1.67173  
 -1.78861   2.32399    0.0359288

In [10]:

inv(A+dA)-inv(A)

Out[10]:

3×3 Array{Float64,2}:
  0.00361557    0.0219253  -0.0380485
 -0.000140811  -0.0161177   0.0188606
  0.00369236    0.0071466  -0.0214998

In [11]:

-AI*dA*AI

Out[11]:

3×3 Array{Float64,2}:
  0.0036997   0.0220909   -0.0385415
 -0.0001703  -0.0161938    0.0190541
  0.0037521   0.00724607  -0.0218293

In [ ]: