(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A square matrix A is called nilpotent if A^k =0 for some k>0. Prove that if A is nilpotent then I+A is invertible.

3. The attempt at a solution

My guess would be to do a proof by induction (on the size of the matrix)

So for the trivial cases:

Let A be a 2x2 Nilpotent matrix... thus it is of the form

[0 x] [0 0]

[0 0] Or [x 0]

Clearly when we add I to A in this case, we get get a matrix, whose det =/= 0

Im having trouble doing the more general cases, seeing as that i cannot mentally see what nilpotent matrices of a larges size look like.

Is this even the best way to approach this problem?

Any help is appreciated.

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# Homework Help: Linear Algebra: Proof involving basic properties

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