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

[mathmathmad]

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?

 

[CompuChip]

I presume it would be the zero matrix (the unique m x n matrix all whose entries are 0).

 

[HallsofIvy]

In other words, if A is an invertible matrix, is it possible to have $A^n= 0$ for some n? Start multiplying both sides of $A^n= 0$ by $A^{-n}$. "Repeat as needed".

 

[CompuChip]

I would choose another approach, where the hint "consider the determinant" applies.

Take your pick :)