Let A be the matrix 1 1 1 0 Compute A2 A3 A4 A5 and A6 What

Let A be the matrix [1 1 -1 0] Compute A^2, A^3, A^4, A^5, and A^6. What will A^7, A^8, and A^9 be? If we compute A^k for k = 1, 2, 3, 4, ..., how many different matrices will we get? Find the characteristic polynomial of A, and compute its (complex!) eigenvalues. Looking at the answers to question (2), explain why A has the behavior above. [Thinking about the fact that there is a formula for the entries of A^k in terms of the eigenvalues of A, or what A would look like in diagonal form is one way to think about (e), but there are others.]

Solution

Dear Studet Thank you for using Chegg !! Given A = 1 1 -1 0 A^2 = 0 1 -1 -1 A^3 = -1 0 0 -1 A^4 = -1 -1 1 0 A^5 = 0 -1 1 1 A^6 = 1 0 => Clearly matrix A has a cyclicity of 6 i.e. A^6 = A 0 1 b) A^7 = 1 1 -1 0 A^8 = 0 1 -1 -1 A^9 = -1 0 0 -1 c) As identified correctly above with increasing powers of A, we shall have 5 more matrices apart from A i.e. in totality we have 6 different matrices in different powers of A d) Character matrix of A A = 1 1 -1 0 A - I = 1 - 1 -1 - Characteristic polynomial is = (1-)(-) + 1 = 0 = ^2 - + 1 = 0 = (1 + 3i) / 2 (Eigen vaues) Kindly note that as per Chegg instructions, an expert can answer at most 4 subparts of a question