I"m pretty certain this would need that $B$ likewise be invertible for it to it is in true.
You are watching: If a and b are invertible then ab is invertible
Is over there a quick means to disprove the if $A$ is invertible climate $AB$ is invertible?
The proof the if $A$ and also $B$ room invertible, climate $AB$ is invertible have the right to be done an ext elegantly if you recognize these two results:
$(1)$. $detAB = (det (A))*(det(B)).$$(2)$. A procession $B$ is invertible if and also only if $det(B) eq 0$.
Proof: expect that both $A$ and $B$ room invertible. Climate $det(A) eq 0$ and also $det(B) eq 0$. Now by $(1)$, $det(AB) eq 0$, for this reason by $(2)$, $AB$ is invertible.
I don"t quite acquire your question since the insurance claim is false. What if $B$ is the zero matrix?
Implication of $(AB)^-1 implies det(A) eq 0$ is trivially true$$(AB)^-1 = C = B^-1A^-1 = C implies A^-1 = BC = B(AB)^-1$$
But the allude is, the converse walk not apply (if $A$ is invertible then $AB$ is invertible).
Consider the square matrices as direct maps $digitalrecordersreview.orgbbR^n o digitalrecordersreview.orgbbR^n$. These three conditions are equivalent: $A$ is invertible, $operatornameim A = digitalrecordersreview.orgbbR^n$, $operatornameker A = 0$.
Since $operatornameim abdominal subseteq operatornameim A$, we have actually $$ operatornameim abdominal = digitalrecordersreview.orgbbR^n Rightarrow operatornameim A = digitalrecordersreview.orgbbR^n. $$Due to the equivalent conditions mentioned above: $$ abdominal ext is invertible Rightarrow A ext is invertible. $$
You can use the 3rd condition to display that $B$ is additionally invertible.
Thanks for contributing solution to digitalrecordersreview.orgematics Stack Exchange!Please be certain to answer the question. Provide details and also share her research!
But avoid …Asking for help, clarification, or responding to various other answers.Making statements based upon opinion; back them increase with recommendations or an individual experience.
Use digitalrecordersreview.orgJax to format equations. digitalrecordersreview.orgJax reference.
See more: She Wants Revenge Written In Blood — She Wants Revenge, She Wants Revenge
To learn more, view our tips on writing an excellent answers.
post Your answer Discard
Not the answer you're looking for? Browse various other questions tagged linear-algebra matrices train station or asking your very own question.
site architecture / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.11.5.40661