Let $A,B\in M_{n\times n}\left(\mathbb{R}\right)$, If $A$ is Not Invertible prove that $AB$ and $BA$ aren't invertibles?

Question

Let $A,B\in M_{n\times n}\left(\mathbb{R}\right)$, If $A$ is Not Invertible prove that $AB$ and $BA$ aren't invertibles?

Proof for $AB$:

Assume by contradiction that $AB$ is invertible, then there is C such that: $\left(AB\right)C=I_{n}\stackrel{\text{Association}}{\iff}A\left(BC\right)=I_{n}$

Which means $A$ is invertible from the right in contradiction to $A$ not being invertible .

I need help with proving for BA

Note. I can't use det and other advanced concepts

Can you not use the exact same argument but left-multiply by the assumed inverse? — Jakob Streipel, Apr 18 '23 at 14:38
Can you explain how it helps? I get that $(CA)B=I_n$ but how it contradicts that $A$ is not invertible? — CStudent, Apr 18 '23 at 14:42
Actually, it does. Next time please look for duplicates before asking. — Anne Bauval, Apr 18 '23 at 14:56

score -2 · Accepted Answer · answered Apr 18 '23 at 14:50

-2

Suppose $BA$ is invertible. Then, there exists a matrix $D$ such that $$(BA)D = D(BA) = I$$ but then, $$(DB)A = I$$ implying that $A$ is left-invertible, a contradiction.

answered Apr 18 '23 at 14:50

stoic-santiago

13,705

1

Please don't rush to answer such a FAQ. Rather search (and vote to close) for (multi!)duplicates. – Anne Bauval Apr 18 '23 at 14:55
@AnneBauval Sure! I'll keep this in mind. Thank you. – stoic-santiago Apr 18 '23 at 18:02

Let $A,B\in M_{n\times n}\left(\mathbb{R}\right)$, If $A$ is Not Invertible prove that $AB$ and $BA$ aren't invertibles?

1 Answers1