Question

4. (-12 points) DETAILS LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) -2 -2 A 0 3-2 0 -1 PE 11 Verify that p-IAP is a diagonal matrix with the eigenvalues on the main diagonal. P-AP Need Help? Read it Talk to a Tutor Submit Answer 5. [-12 Points] DETAILS LARLINALG8 7.2.013. For the matrix A, find (if possible) a nonsingular matrix P such that p-AP is diagonal. (If not possible, enter IMPOSSIBLE.) 1 0 52-5 -1 0 0 2 Verify that p-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-IAP =

Answer 1

characteristic polynomialo: 2-1 -2 3 0 3-1 -2 0 -1 2-2 = -23+722 - 142 + 8 = - (1 - 1)*(22 - 61+8)= -(1-1)×(1-2) (1-4) 1.21 = 1 2.12 = 2 3.13 = 4 2. For every 1 we find its own vector(s): 1.21 = 1 1 -2 3 A-2 11 = 0 2 -2 0 -1 1 Αν = λν (A-2) = 0

-X3 General Solution : X = X3 X3 The solution set: {* 3 * -1 1 1 efort (1) III Let x 3 = 1, V1 = 2.22 = 2 0-2 3 A-121 = 10 0 1 -2 0 -1 0 Αν = λν (A - XI) v = 0

General Solution : X = (3) for-18) The solution set: {x1 Let X 1 = 1, V2 = O 1 0 3.13 = 4 -2 -2 3 A-131= 0 -1 -2 0 -1 -2 Αν = λν (A-AI) = 0

General Solution : 7 3 3x3 X = -2X 3 ( X 3 The solution set:{X3 7 2 -2 Х - 1 7 2 Let X 3 = 1, V3 = -2 1

3. • The diagonal matrix (the diagonal entries are the eigenvalues - 11,12,13): 1 0 0 D = 0 2 0 0 0 4 4) . The matrix with the eigenvectors (V1, V2, V3) as its columns: 7 -1 1 2 P = 1 0 -2 1 0 1 = 0 p(-1) = 1 3 3 2 -1 3 2 3 -1 2 1 3 0

1. varification PIA - o 23 2 -2 3 7/27 3/2 12 0 - 2 3 2 2 1 0 • YB 113 ² 23 H2 3 -1 0 -2 <4/3 - Al3 O 2 o with the el this shows that prAP is a diagonal onal matrix on the main diagonal. esgen ratus