###### Exercise 3

Suppose that \(A\) is an \(n\times n\) matrix.

Suppose that \(A^2 = AA\) is invertible with inverse \(B\text{.}\) This means that \(BA^2 = BAA = I\text{.}\) Explain why \(A\) must be invertible with inverse \(BA\text{.}\)

Suppose that \(A^{100}\) is invertible with inverse \(B\text{.}\) Explain why \(A\) is invertible. What is \(A^{-1}\) in terms of \(A\) and \(B\text{?}\)