A = B+ I, B^2 = 0

Let ${A = \begin{pmatrix} a + 1& -a\\ a & -a + 1\\ \end{pmatrix}}$. What can we say about ${A}$?

Remark: ${A}$ has some interesting properties: ${A = B + I}$ where ${B^2 = O}$.

1. ${A^n = nB+I, n \in \mathbb{Z}}$.
Indeed, we can prove it by induction.
${A^2 = (B+I)(B+I) = B^2 + 2B + I = 2B+I}$, ${(B+I)(-B+I) = I \Rightarrow A^{-1} = -B + I, A^{-2} = (-B + I)(-B+I) = -2B+I\dots}$ Suppose that ${A^k = kB+I}$, we have ${A^{k+1} = (kB+ I)(B+I) = kB^2 + (k+1)B + I = (k+1)B + I}$. Moreover, suppose that ${A^{-k} = -B+I}$, we have ${A^{-k-1} = (-kB+I)(-B+I) = -(k+1)B + I}$.
2. ${A^{-1} = - B+I \Leftrightarrow B + I = - A^{-1} + 2I \Leftrightarrow A = -A^{-1} + 2I \Leftrightarrow A + A^{-1} = 2I}$.