<s>[INST] If $A+B = I_n$ and $A^2 +B^2 = O_n$ then $A$ and $B$ are invertible and $(A^{-1}+B^{-1})^n = 2^n I_n$ For $A$ and $B$ square matrices of size $n$, show that:
If $A+B = I_n$ and $A^2 +B^2 = O_n$ then $A$ and $B$ are invertible and $(A^{-1}+B^{-1})^n = 2^n I_n$
Where $I_n$ is the identity matrix of order $n$, and $O_n$ is the square matrix of size $n$ with all the entries equal to zero. 
 [/INST] First, let's show that A is invertible:
$$\begin{align} A+B=I\\B=I-A\\A^2+(I-A)^2=0\\I-2A+2A^2=0\\A(2(I-A))=I\\A^{-1}=2(I-A)\end{align}$$
Similarly, $B^{-1}=2(I-B)$, so $A^{-1}+B^{-1}=2(2I-A-B)=2I$ (because $A+B=I$), so:
$$(A^{-1}+B^{-1})^n=(2I)^n=2^nI$$.
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