Linear Algebra Questions

1. Consider the matrix A = 12 a. Determine the eigenvalues A and find the corresponding eigenvectors for each eigenvalue. b. Describe what the matrix A is doing as a transformation on R by describing its actions on the eigenvector directions. 2. Suppose a matrix A has characteristic polynomial p(1) = 1(1 - 2) (1 - 1)3 (2 + 2) 3. a. What is the size of A? b. Is A invertible? c. How many eigenspaces does A have? d. For each of these eigenspace, tell me what the dimension is or (if it impossible to know the exact dimension), tell me the possible range of values for that dimension. e. If the eigenspace for 1 = -2 has dimension 2, then will A be diagonalizable? 3. Consider the matrix A = [8 4] a. Determine the eigenvalues of A. b. For each eigenvalue in a., determine a basis for the corresponding eigenspace. c. Is A diagonalizable? If so find an invertible matrix P and diagonal matrix D such that P-AP = D. (You should not have to find P-1 to do this!)