I need this question answered please Consider a square matri

I need this question answered please.

Consider a square matrix A. a. What is the relationship among ken A and kee (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), ...? b) What can you say about im(A), im(A^2), im(A^3), ...?

Solution

Let x be an arbitrary vector in Ker(A) . Then Ax = 0 . Further, A2x =( A.A)x = a(Ax) = A.0 = 0. Therefore x Ker (A2) Hence Ker (A) Ker (A2) . The converse is not necessarily true as Ker (A2) may contain other vectors also. On extending this argument further, we observe that Ker (A) Ker (A2) Ker (A3) Ker (A4)….. We know that Im (A) is same as Col(A). Therefore, any vector x, A2x is in Im(A2). Further A2x = (A.A)x = A(Ax) = Ay (say) where Ax = y. We know that Ay Im(A) . Therefore A2 x     Im(A). This implies that Im(A2) Im(A). On extending this logic further, we have, Im(A4) Im(A3) Im(A2) Im(A) and so on.