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## 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.