Question

2. [-12 Points) DETAILS LARLINALG8 7.2.005. Consider the following. -4 20 0 1 -3 A = 040 P= 04 0 4 0 2 1 2 2 (a) Verify that A is diagonalizable by computing p-AP. p-1AP = 11 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (91, 12, 13) = Need Help? Read it Talk to a Tutor 3. [-12 Points) DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that P-AP is diagonal. (If not possible, enter IMPOSSIBLE.) A = -4 P= 11 Verify that p-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP = 11

Answer 1

2. Given that 0 A = -4 2 0 4 4 0 0 0 4 2. 2 1 2 now compute inverse matrix of p re.pt To find pl, we augment with the identity matrix and using now operations we make identity matrix to the left. Then to the night will be inverse matrix pl RR, 0 1 - 3 [PII] = 10 1 o O 1 2 2 1 4 o 0 1 0 4 O 1 2 2 0 1 O 1 1 0 0 Rg x 1 4 0 2 0 -42 1 1. 2 2 0 1 0 R-2R2 0 0 0 1 0 0 0 -3 14 -1/4 "R3-R2 0 1/40 1 0 0 1 O 1 -3 0 1 O 2 0 -Y2 1 R, - 2 R3 1 0 o -H3 23 114 1 1 a 0 114 0 0 1 0 0 0 0 9/12 0 0 1 -/ 9/12 a we have oblained the identity matris to the left. So we are done. so, 2 O - 4. Р 914 O -418 1/12 0 Now, P'AP- - 23 ܝܐ 2. 0 0 1 - 3 0 1/4 0 4 0 4 1-Y3 1/12 0 4 2 1 2 2. H2 -H9 1 4 12 16 0 0 0 2 8 -8 -73 1/12 0 2 O 0 0 4 0 0-4 1 since phap is diagonal matrix then we say that A is diagonalizable.

let Then B - 2 4 is a 0 (6) B = plap A, B are similar matrices Again, diagonal matrix. So its --4 eigenvalues are its diagonal elements 2. A2 = 4, A2 = 4, 03 = -4 Since A, B are similar matricen then they have the same eigenvalues So, (A, , 22, 23) = ([2,4,-4) il. A
Using the concept of diagonalization of a matrix and by given theorem I solve the problem .