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Let $A$ be an $n \times n$ matrix with real or complex entries and such that $A^3=0.$ Which of the following options holds?
1. $(I+A)^3=0$.
2. $I+A$ is invertible.
3. $I+A$ is not invertible.
4. Necessarily $A=0$.

Can someone point me in the right direction? Thanks in advance for your time.

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2 Answers 2

Hint:

$$(I-A)(I+A+A^2)=I-A^3\ldots$$

Think it slowly. Well chewed this hint solves all first three questions. The fourth one is almost trivial.

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up vote 0 down vote accepted

Using DonAntonio's suggestion,we see that $(I+A)(I-A+A^2)=I+A^3=I$ ,since $A^3=0.$ This gives $|(I+A)(I-A+A^2)|=1 \implies |(I+A)||(I-A+A^2)|=1 \implies |(I+A)| \neq 0.$ So option 2 seems correct.

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I think you're supposed to determine whether each holds in turn. It's possible more than one does or that none do. –  Potato Mar 29 '13 at 4:49
    
option $(2)$ holds $\implies (3)$ does not hold. The option $(4)$ is trivial. But I am not sure about option $1$. Can you help? –  learner Mar 29 '13 at 4:55
    
Take $A$ to be the zero matrix. –  Potato Mar 29 '13 at 4:56
    
Thanks.Then option $(1)$ is clearly false. –  learner Mar 29 '13 at 4:58
1  
Indeed. $ $ $ $ –  Potato Mar 29 '13 at 4:58

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