Find the Eigen values and corresponding linearly independent Eigen vectors.1.4 6 61 3 21 4 3       

(a) Find the eigenvalues, eigenvectors and eigenspaces forA =1 0 53 6 02 0 10

a) Find the eigenvalues and the associated eigenvectors of the matrixA = [7 0 −3−9 −2 318 0 −8]

To find the eigenvalues, computedet3 − λ 0 0−3 4 − λ 90 0 3 − λ = (3 − λ)(4 − λ)(3 − λ).So the eigenvalues are λ = 3 and λ = 4.We can find two linearly independent eigenvectors301 ,130 corresponding to the eigenvalue 3, and oneeigenvector010 with eigenvalue 4. The diagonalized form of the matrix is3 0 0−3 4 90 0 3 =3 1 00 3 11 0 03 0 00 3 00 0 40 0 11 0 −3−3 1 9 .Note that if you chose different eigenvectors, your matrices will be different. The middle matrix should haveentries 3, 3, 4 in some order, and you should multiply out the product to make sure you have the right answer.

(a) Find the eigenvalues, eigenvectors and eigenspaces forA =1 0 53 6 02 0 10(b) Calculate the determinant and trace of A directly and using eigen-values.

Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix.−1 32  − 121