## Acalytica

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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.
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