# Linear Algebra- Kernel and images of a matrix

## Homework Statement

Consider a square matrix A:

a. What is the relationship between ker(A) and ker(A^2)? Are they necessarily equal? Is one of them necessarily contained in the other? More generally, What can you say about ker(A), ker(A^2), ker(A^3), ker(A^4),...?

b. What can you say about im(A), im(A^2), im(A^3), im(A^4),...?

2. The attempt at a solution

So i believe if A is invertible nxn matrix, than ker(A)={<0,0,0>} and so will ker(A^2) and so on. And the image of A if A is invertible is im(A)=R^n and so will the im(A^2) and so on, but im not sure what it would be for other conditions of A at least thats what I think this question wants.

a. If $$x \in \ker(A)$$, what can you say about $$A^2 x$$? What does that tell you about $$\ker(A^2)$$?

b. For any $$x$$, $$A^2x$$ is in $$\operatorname{im}(A^2)$$. If you rewrite $$A^2x$$ as $$A(Ax)=Ay$$, what can you say about $$\operatorname{im}(A)$$?

for A: A^2 * x would equal the zero vector which means the ker(A^2) contains x but is not necessarily equal because it may contain another vector

B. Is that saying the im(A) contains the im(A^2) but is not necessarily equal to the im(A^2) because it will span more vectors?

for A: A^2 * x would equal the zero vector which means the ker(A^2) contains x but is not necessarily equal because it may contain another vector

In other words, we have $$\ker(A)\subset \ker(A^2)$$.

B. Is that saying the im(A) contains the im(A^2) but is not necessarily equal to the im(A^2) because it will span more vectors?

Yes. You could also write : $$\operatorname{im}(A^2)\subset \operatorname{im}(A)$$

Now, how can you apply this to $$\ker(A^3)$$, $$\ker(A^4)$$... and $$\operatorname{im}(A^3)$$, $$\operatorname{im}(A^4)$$...?

alright, thank you very much for your help