Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) [14] λ; = 4,4 X For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x;) =

4. (a) Using Cayley-Hamilton theorem find the inverse of the matrix [i 2 1] A =| 2 1 0 1 1

Determine the eigenspaces of the matrix A. +-649 1 1 1 1 -1 2, A

Show your reasoning for each exercise. While discussions among students concerning the course material and homeworks are allowed, each student must write and submit their own, independent solutions. Let T be the 2-by-2 matrix with real coefficients. Everything below can be generalized to a general 2-by-2 matrix, to a general n-by-n matrix, and even beyond, but we want to keep computations tractable. Recall that a complex number A is called an eigenvalue of T if there exists a nonzero vector v such that Τυ = λυ. Let us call o the set of all the eigenvalues of T. Note that X E o if and only if the matrix -b λ-a 0 X-d, is not invertible. This last condition is in turn equivalent to 0 = det(X-T). 1. Compute det (C – T) as a function of and list all possibilities for the number of (complex) zeros this function has i.e. eigenvalues that the matrix T has. Show that this inverse has matrix form by computing the product X-T= Hint: Recall our discussion on the multiplicity of zeros, or compute…