Is A^0 Equal to the Zero Matrix O_m,n?

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Homework Help Overview

The discussion revolves around the properties of matrix exponentiation, specifically whether raising an invertible matrix A to the power of zero results in the zero matrix O_m,n. The subject area includes linear algebra and matrix theory.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants explore the definition of the zero matrix and question the implications of matrix exponentiation for invertible matrices. Some suggest examining the determinant as a potential approach to understanding the problem.

Discussion Status

The discussion is active, with participants offering different perspectives and hints for approaching the problem. There is no explicit consensus yet, but various lines of reasoning are being explored.

Contextual Notes

Participants are considering the implications of the properties of invertible matrices and the definitions involved, such as what constitutes the zero matrix.

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Homework Statement



state if the statement if it's true or false (give a brief explanation if it's true or give a counter example if it's false)

Homework Equations


If A is an invertible nxn matrix, then A^m is not equals to O_m,n for all natural m


The Attempt at a Solution


what is O_m,n?
 
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I presume it would be the zero matrix (the unique m x n matrix all whose entries are 0).
 
In other words, if A is an invertible matrix, is it possible to have [itex]A^n= 0[/itex] for some n? Start multiplying both sides of [itex]A^n= 0[/itex] by [itex]A^{-n}[/itex]. "Repeat as needed".
 
I would choose another approach, where the hint "consider the determinant" applies.

Take your pick :)
 

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