Exercise 6

Suppose that $A$ is an $n\times n$ matrix.

1. Suppose that $A^2 = AA$ is invertible with inverse $B$. This means that $A^2B = AAB = I$. Explain why $A$ must be invertible with inverse $AB$.

2. Suppose that $A^{100}$ is invertible with inverse $B$. Explain why $A$ is invertible. What is $A^{-1}$ in terms of $A$ and $B$?