Question

2. (-12 points) DETAILS LARLINALG8 7.2.005. Consider the following. Toi-3 A = 5 - 1 0 0 1 0 -6 4 - 4 P= 04 1 2 0 2 (a) Verify that A is diagonalizable by computing p-1AP. p-1AP = It (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. (11, 12, 13) =

Answer 1

Solution- Diagonali zability of a Matrix A :- let Anxo be any matrix then A is said to be Diagonalizable over F if AND (A is similar to a Diagonal Matrix D). - 5 -1 O PE 0 A - 0 0 0 0 4 - 64 -4 es es @ Compute P-AP First we have to compute p- & We know that p = I adjp IPI 1P) = 018-0)-0(2+6) + 1 (0+12) = 12 ( By Expanding determinant, by C, ) adj P = P. Pig Pig where Pij denotes cofactor of li, j) P2, P22 P23 Pij Minor at ligj) Ps3 where [ J' this represents transpose of the matrix 1 (8-0) 8 (-1) = = = Pri = (1) 61)'*? Pia 11 -1 (0-0) = 0 20 Similarly o co P 13 = -4 P31 = 12 o adj P: - 3 -8 11 = -8 12 0 0 = 0 Pas = 3 P3} '8 12 13 -8 TO 3 0 3 0 0 P23 = 1 26 " -4 10 -

To find : PAP Now PHAP = (PAP (because Matrix Multiplications So, first we hawe to find plA. is ayociative) -8 12 PTA = I 5 O 0 0 3 -6 10 4 -4 -4 -32 32 -48 11 lyrex matrix Multiplication) १२ 0 3 0 o -20 5 0 -48 0 0 - -32 32 - 48 HAP= 1 13. 4 0 0 0 ।२ 3 21 0 0 60 - 2 or -20 s -4 0 0 0 - 0 0 5 & We know that A & PPAP are similar it 1P# → A1 PHAP + AND ( a Diagonal Matrix) → Ais Diagonalizable

③ Result :- If A & B are similar nxn matrices then they have same eigen values. Since By part @ AND then A and I have some eigen values & We know that for a Diagonal Matrix the eigen values of that Diagonal matrix lies on the Diagonal entries of the Diagonal Matrix. : Eigen Values of I are -4, 1,5

.: Eigen values of a wre -4,1,5 also. ( because A & Dare similax matrica) .. A&D have same eigen Values