Proof of definition of invertible matrices

Question

Let $A \in \Bbb R^{n \times n}$. Then $A$ is invertable if and only if a Matrix $B \in \Bbb R^{n \times n}$ exist such that $AB=E$.

This seems like the definition of an invertible matrix but how do you prove the definition? As a physics student I don't do many proofs so it would be nice if someone could give me hint.

You cannot prove definitions. Do you rather want to show that $AB=E$ implies that $BA=E$ as well? — Hagen von Eitzen, Jun 15 '15 at 21:01
That's why it is not exactly the definition of an inventive matrix... — mathcounterexamples.net, Jun 15 '15 at 21:02
Then pray tell what is the definition of invertible matrix? — Hagen von Eitzen, Jun 15 '15 at 21:03
In linear algebra, an $n \times n$ square matrix $A$ is called invertible if there exists an $n \times n$ square matrix $B$ such that: $$AB=BA=I$$
@HagenvonEitzen I don't even know what I want to show. — qmd, Jun 15 '15 at 21:04
@Jean-PierreMerx So the definition in my opening post is incomplete? Then how can I prove that statement is true? — qmd, Jun 15 '15 at 21:07
Yes it is incomplete. SuH gave the precise one in his comment. — mathcounterexamples.net, Jun 15 '15 at 21:08
@Jean-PierreMerx SuH is me ;). This is exactly what confuses me. How can I prove an incomplete definitiont? Doesn't make any sense to me. — qmd, Jun 15 '15 at 21:10
You need to show that the existence of $B$ with $AB=I$ implies that $BA=I$. My approach would be: $AB=I\Rightarrow det(A)\neq 0 \Rightarrow A$ is invertible. — Taylor, Jun 15 '15 at 21:13
This may be relevant: http://math.stackexchange.com/a/3860/27978. — copper.hat, Jun 15 '15 at 21:15
@copper.hat This combined with Taylor's hint is very helpful. Thanks! — qmd, Jun 15 '15 at 21:19
I don't know why this was put on hold as unclear, I know exactly what it's asking. — , Jun 16 '15 at 00:38

score 2 · Answer 1 · answered Jun 15 '15 at 21:59

2

Hint: $AB=I \implies B(AB)=B \implies (BA-I)B=0$

answered Jun 15 '15 at 21:59

karakfa

2,675

Proof of definition of invertible matrices

1 Answers1