Question

3. [-12 Points] DETAILS LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) 8 -2 A= P= Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP =
please solve both

Answer 1

8 - 2 A -4 first we find eigen values of A. det IA-XI) 2o. 8- -2 zo 4 (8-))(1-2) – (8) 20. 8-80-ta2 -8=0 AM ²_9x=0 A(4-9)=0 a=og (A -AI)x = Eigen vector also for A=0 =0 [*2.J[% 3303) s =0 4 R₂ » 2 R2 + RI 8 % -27 -0. -2 8 2] 3] 47 -Y Eigen vector for X = k, (let) 4.x = y Y = 4K, 120. (7) - k[ ]

for X-9 (A ->I) x 20 8-9 - 2 (2 23][2] -4 1-9 - 2 al co 4 -8 y R₂ → R2-4R, - x - 2y =o. = ) E-3) (0,33 Y & = K₂ (let) 2y = -x. Eigen vector for Х a=9 kz (-3 -2 P = و ] Answer. IPI 1 - (-8) = 9. 2 e- - IPI [P ?] = $ 1-4 :] -4

2 PARS [:] [4"}][: 112 8 -8 - 16 2 - 2 P'AP [-4 9 - 4 - 4 +4 8 + 1 -18 -1] plap -18 +18 pt HAP 9 0 +72+9 5[4] [. [ Ś le to ;] [2] o - phap 81 plap verify whenre d=0 az =9 plap Diagonal matrix,

Ques - -۱۱ -4 -5 A = [" - م [ ] - 2 + pla AP Diagonal Matrix with eigen values oot are the diagonal element. . 4 5 4 - (5) (s) = - (2) = 1 -5 - لام - [ رح | -4 4+ا۔ plap و4 \\۔ | ا۔ ة - و4 - 44 و4- کد + ]: [ -- ]:[ = [: ][ ][- ] [ ] [ - +] [ ]-[ ۱۹۴۰ = = =[ 2] = * - و 4 کا ک- 4 گا۔ کا || 3 م ه - 12 + گا- plap. where کم pt AP ره بلا Answer eigen values 0 -3 || - . 3 - = م (۱۰ ) : (-1,-3). Answer