<s>[INST] Determine what condition the $n\times n$ matrix $A$ must fulfill so that the matrix $I_n - A$ has an inverse and this equals $I_n + A$. 
Determine what condition the $n\times n$ matrix $A$ must fulfill so that the matrix $I_n - A$ has an inverse and this equals $I_n + A$.

I tried to apply it to a $2\times 2$ matrix obtaining the following result:
$A =
\begin{pmatrix}
    a & b\\
    c & d
  \end{pmatrix}; $
$(I_n-A)^{-1} = \frac{1}{(1-a)(1-d)-bc} *
\begin{pmatrix}
    1-d & b\\
    c & 1-a
  \end{pmatrix} =
\begin{pmatrix}
    1+a & b\\
    c & 1+d
  \end{pmatrix} $
 [/INST] $(I - A)^{-1} = I + A \Longleftrightarrow A^2 = 0; \tag 1$
for if
$A^2 = 0, \tag 2$
then
$(I - A)(I + A) = I + A - A - A^2 = I - A^2 = I; \tag 3$
likewise if
$(I - A)(I + A) = I, \tag 4$
then
$I - A^2 = (I - A)(I + A) = I, \tag 5$
then
$I = A^2 + I, \tag 6$
or
$A^2 = 0. \tag 7$
</s>