Explicit expression for inverse of I-A

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SUMMARY

The discussion centers on finding an explicit expression for the inverse of the matrix (I - A), where A is a square matrix (n x n) and A^k = 0. It is established that the inverse can be expressed as (I + A + A^2 + ... + A^(k-1)). However, participants agree that no further simplification or alternative expression exists without additional information about matrix A. The conclusion is that the provided expression is indeed the most explicit form available under the given conditions.

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  • Familiarity with the properties of nilpotent matrices (A^k = 0).
  • Knowledge of series expansion in matrix contexts.
  • Basic concepts of linear algebra, particularly square matrices.
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peripatein
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Hello,
This is not a homework exercise, so I decided to post it here. Hopefully one of you could help.
I would like to find an explicit expression for (I-A)^(-1), provided that A is a squared matrix (nxn) and A^k = 0. It is also given that I-A^k = (I-A)(I+A+A^2+...+A^(k-1)).
I understand that by definition the inverse matrix of I-A will be (I+A+A^2+...+A^(k-1)), but is there a way to arrive at a more simplified, explicit expression (yet without knowing what A is)?
 
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I don't understand why

[tex]I+A+A^2+...+A^{k-1}[/tex]

isn't good enough for you. It's an explicit expression.

To my knowledge, there is no other expression.
 

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