Question

2. For the following matrices, find a matrix Pthat diagonalizes A and compute p-'AP. If the matrix is not diagonalizable, say so and give a reason why. -14 12 (a) A= -2017 100 (b) A011 011

Answer 1

12 (-14 2.(a) A = det AF 14x17 – 12 (20) 2 -20 17 A is Characteristic I sum of diagonal element] equation of - (trace A) & det A = 0 ² – 3x + 2 = 0 trace = (x-1) (x-2) = 0 x= 1, 2 let x=1, x2 = 2 to eigen * Let X= be vector corresponding eigen value 1-15 12 Ax = xix » (A - x, I) x = 0 where A-X, I = 20 16 equations, we system of о We have solve linear ellimination it by homogeneous Gaussian - 1/3XR, - 1 -4 4/5 -15 120 R2 +20P, 10) 20 16 0 16 - 20 . * = 4 x 2 X

be eigen vector corresponding to eigen A2 = 2 -16 12 homogeneous system of Let value AX = 2x - (A-d, I)x=0 where This is and solve it by 1-16 - XR, - (A-X, I) -20 15 linear equation Elimination Gaussian 0 12 1 R2+20R, 0 - 20 15 -20 15 O x - x = 22 as its columns the eigenvectors The matrix with is P = - is where PAP =D where diagonal matrix 0 O A 1 - - Characteristic equation of A 0 I-X 1 = 0 0 I- - → (1-x) {(1-x)(1-4)-1} = 0 (1-x) (1 - 2x + x - 1) = 0 (1-x) (-2x) = 0 (1-7)(x-2)=0 x = 1, 0,2 Let x=0, A2=1, +3=2

Let X = to 33 be eigen vector eigen value corresponding x = 0 ОО A-X, I = 1 0 0 AX = XIX = (A-X, I) x=0 where There homogeneous system of solve it by Gaussian linear equations, is a Elimination we 1 O O o 1 0 0 R3-R₂ O o :: 13 o 1 0 0 0 X2=-23 22 0 X = Now foo 12 = 1 O 0 AX - X, X - (A-7₂ I) x=0 where A-X₂ I = O 0 0 0 o xz x = (3) - (0) Now for x3 = 2 A X = X 3x - (A-X3I) x=0 -1 0 *650 0 1 -1 -9, = 0 -> 2,0 - X2 taz o x2 =23 X2 - 23 =0 x2 =23

O X= The its columns matrix as with the eigen vectors O 0 - P = O 1 0 1 where ооо PAP =D D 0 0 O 0 2 1 is a diagonal matsin.
Both the matrices are diagonalizable.