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Let $A, B, C$ be $n \times n$ matrices and assume $AB = C$. Prove if $C$ is invertible, then $A$ and $B$ are invertible and specify inverses.

  • See also https://math.stackexchange.com/questions/1729990/prove-that-if-a-and-b-are-square-matrices-and-ab-is-invertible-then-both?rq=1 – Arnaud D. Mar 08 '18 at 11:50

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Hint: Let $D$ be the inverse of $C$. Then $CD=DC=I$. Now multiply $AB=C$ by $D$ separately on the right and on the left.

If you don't need to give the inverses, just compute the determinant on both sides.

lhf
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  • Is this true in the infinite-dimensional case? (I mean, the fact that $A$ and $B$ are invertible if their product is) – Giuseppe Negro Mar 08 '18 at 11:52
  • @Giuseppe It appears that this proof works for functions in general whose composition is invertible, I would expect it to apply to the infinite dimensional case – pjs36 Mar 08 '18 at 12:16
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    @pjs36 this is definitely wrong in the infinite dimensional case. The argument only shows that $A$ has a right inverse (which corresponds to "surjective") and $B$ has a left inverse (which corresponds to "injective"). But only in the finite dimensional case this is enough to conclude that $A$ and $B$ are invertible. – Claudius Mar 08 '18 at 12:29