## Related questions with answers

Question

Let A and B be $n \times n$ matrices, and suppose that B and AB are both invertible. Prove that A is also invertible.

Solution

VerifiedSince $B$ is invertible, so $B^{-1}$ is also invertible by Theorem 3.23 part (a). Now $(AB)B^{-1} = A(BB^{-1}) = AI = A$, so we have $A = (AB)B^{-1}$. Since both $AB$ and $B^{-1}$ are invertible, so $A$ is product of two invertible matrix. Therefore by Theorem 3.23 part (b), $A$ is invertible.

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