The question states:
"Consider a square matrix $A$ with $\ker(A^2) = \ker(A^3)$. Is $\ker(A^3)= \ker(A^4)$? Justify your answer."
It's clear that if $x$ satisfies $A^3x = 0$, then it satisfies $A^4x = 0$. I am wondering whether the opposite is true.
Equal nullspaces of consecutive powers of a matrix
1
$\begingroup$
matrices
1 Answers
3
Suppose $A^4x=0$, then $A^3Ax=0$. Hence $Ax\in \ker(A^3)=\ker(A^2)$. Thus $A^2(Ax)=0.$ Thus $A^3x=0$. This shows that $\ker(A^4)\subset \ker(A^3)$.
In fact $\ker(A^n)=\ker(A^2)$ for all $n\geq 2$ under these conditions.