Please Include all steps formulas, scratch work and identities used to solve i am looking to learn the approach mostly. Thank you

2. Fill in the four gaps in the proof below. (4 pts) Claim. If A is an n x n square matrix, then ker(A2) C ker(A3). Proof. Recall that given sets S and T, we say that S C T if every element of S is also an element of T. Now, given a E ker(A2), then A2x = So ABT = A(_ T) = A(_) =0. Thus TE _ ( A 3 ) . Consequently, ker(A2) C ker(A3)