Generate a matrix A by setting B = [-1, -1; 1, 1], A = [zeros(2), eye(2); eye(2),

Question:

Generate a matrix A by setting
B = [-1, -1; 1, 1], A = [zeros(2), eye(2); eye(2), B]
(a) The matrix A should have eigenvalues λ1 = 1 and λ2 = -1. Use MATLAB to verify this by computing the reduced row echelon forms of A - I and A + I. What are the dimensions of the eigenspaces of λ1 and λ2?
(b) It is easily seen that trace(A) = 0 and det(A) = 1. Verify these results using MATLAB. Use the values of the trace and determinant to prove that 1 and -1 are actually both double eigenvalues. Is A defective? Explain.
(c) Compute the rank of X. Are the computed eigenvectors linearly independent? Use MATLAB to compute X-1AX. Does the computed matrix X diagonalize A?