Exercise 2.7.6

Question

Prove that if the product $AB$ of two $n 	imes n$ matrices is invertible, then both $A$ and $B$ are invertible. Do not use determinant for this problem.

Jing Huang · Accepted Answer

\begin{proof}
$AB$ is invertible, rank$(AB) = n$. From Problem 7.5, we have rank$(AB) = n \leqslant$ rank$(A) \leqslant n$. Thus rank$(A) = n$. So is $B$. $A, B$ have full rank and are invertible. 
\end{proof}

Exercise 2.7.6

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Exercise 2.7.6

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