Consider matrix (3 1 1 A = 4 2 E C3,3 3 3 a) Find the characteristic polynomial of A. b) values. Find the spectrum of A and algebraic and geometric multiplicities of individual eigen- c) Find the eigenspace corresponding to each eigenvalue. 11.3 IC

Find all eigenvalues and a basis for each eigenspace of A. Then use them to find values of lambda and phi such that A can be expressed as the product of a scaling matrix with factor lambda and a rotation matrix with angle phi. A = 3    -4 4     3

Find bases for the eigenspaces associated with each eigenvalue of the matrix 3 0 0 -2 3-2 2 0 5 Find, if possible, a nonsingular matrix P such that P-¹AP is diagonal. A =

Is d=2 eigen value 3. an of 3.