Set B = [-1, -1; 1, 1] and A = [zeros(2), eye(2); eye(2), B] and verify that

Question:

Set
B = [-1, -1; 1, 1] and A = [zeros(2), eye(2); eye(2), B]
and verify that B2 = O.
(a) Use MATLAB to compute A2, A4, A6, and A8. Make a conjecture as to what the block form of A2k will be in terms of the submatrices I, O, and B. Use mathematical induction to prove that your conjecture is true for any positive integer k.
(b) Use MATLAB to compute A3, A5, A7, and A9. Make a conjecture as to what the block form of A2k-1 will be in terms of the submatrices I, O, and B. Prove your conjecture.