Question

Determine whether A is diagonalizable. 2 0 2 A = 0 2 2 2 2 0 Yes No Find an invertible matrix P and a diagonal matrix D such that p-1AP = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list. If A is not diagonalizable, enter NO SOLUTION.) (D, P) =

Answer 1

A " 0 2 02. 2 20 too A to must be ral and non ropeated. be diagonalizada its eigen values So is characteri polynominal A [ 5 6 ર ૨ 220 10 2 0 2-0 2 2 2 TA/= (2-2) [-21+ 2y + 2 (-4+2] (2-1) [²24-47 + 4 [-2] (2-2) { [20-4 -4 (A-72 12-21-8 2 0 d²-4x+2 A-8 2 0 dC d. 4 + 2 (1.4) d = 4,-2. 80 Eigen valves are -2, 2, and so A is diagonalizable matrix, Yes

PAP=D where Pis invertible Ppt = I and Dis diagonal Matrix. So D could be witten as scaling makis also somebimeba PAP PP-AP = PD AP = PD. het 41 pmvalues of A det (PAP) - det (8) det As det 0 - det (A) because det 6-1) detip del I) - detto det (A) = Product of Eigen values d, dan so diugenal maboix D NO o -1 o M 1 -| 1 -) - 0 Along first row = 1 [ (0x+) - (x)] - -0 [f(x-1)- ox]] +0. 1 [-1] +1 (1-0 - 3+1 11 Along first oluran I [ @x-1) - (1x))] - (-) [ {1x) - 40770 1 (-13 + 1 (1) Z 1+1

Sno -sno 0 و زير) sino and Column Along tomo. [leso. coo (-sno sno)] tano [cesotho] -tano