self doubt

Question

if $A^{3} = 0$ can we conclude that $A^{2} = A$ that will be valid for zero matrix

Answer 1

let assume modA ≠ 0.so A^-1 exists.so

A^-1 A³=A^-1 0 => A² =0

now A^-1 A²=A^-1 0 => A=0

this contradiction says that modA definitely be 0.

 But from this we can't say A² =A