Question

Let \(A, B\), and \(C\) be \(n \times n\) matrices.

a) If \(A^{2}\) is invertible, show that \(A\) is invertible.
[NOTE: You cannot naively use the formula \((A B)^{-1}=B^{-1} A^{-1}\) because it presumes you already know that both \(A\) and \(B\) are invertible. For non-square matrices, it is possible for \(A B\) to be invertible while neither \(A\) nor \(B\) are (see the last part of the previous Problem 41).]
b) Generalization. If \(A B\) is invertible, show that both \(A\) and \(B\) are invertible.
If \(A B C\) is invertible, show that \(A, B\), and \(C\) are also invertible.