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Suppose $n > 0$ is fixed, and suppose $S$ is a finite collection of $n × n$ matrices with the following property:

if $A, B ∈ S$, then (1) $AB + A + B ∈ S$.

Show that there exists a matrix $A ∈ S$ such that (2) $A^2 + A = 0$.

My first attempt was BWOC so if (2) is false for all $A$ in $S$ then (1) is false since there would be an A in S with (1) not holding , but I'm struggling with showing it with using products of matrices ..as in $BXA$ ..maybe bwoc is the wrong attack ?

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Randino Andantino mozartino
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  • Please don't make an abusive use of acronyms ; I have been obliged to look in a dictionnary for BWOC = By Way Of Contradiction. Think to all people who are not fluent in English... – Jean Marie Apr 14 '20 at 06:50
  • I have taken the liberty to change your title "Showing a powerful result in Linear Algebra" which did not convey the content of your question. – Jean Marie Apr 14 '20 at 07:01

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Define the binary operation $\odot$ on $S$: $$A \odot B = AB + A + B = (A + I)(B + I) - I.$$ This operation is associative, making it a finite semigroup. Now use the fact that every finite semigroup has an idempotent element to conclude the existence of some $A \in S$ such that $$A \odot A = A \implies A^2 + 2A = A \implies A^2 + A = 0.$$

user771918
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