Question

Suppose A is a 3 x 3 matrix with the following eigenspaces. E__ = span (003 Ez = span {O a) Find an invertible matrix P and diagonal matrix D such that A = PDP-1 b) Use this information to find AS.

Answer 1

3x3 matrix with the following Suppose A is eigenspaces 34(1:13) ---/18 버 so x = -1, +2=-1 be two eigen value of A and corresponding eigen vectors are 0 2 1: 0-- Again, &z=2 be an eigen value of A and cornesponding Eigen vector is (a). the diagonal matrix Da ОО Then ob 0 o 13 ral = o To 2 The matrix pa consiting 0 three 2 0 eigen rectors respectively Since det (P) = 0 170 ,then P is non-singular Then pis invertible.

Now + (-2) T - adj P= + (1) (o) (0) -(2) + (1) - (0) ~() + (1) 2. 1 ~2 1 ( 1 2. P- adj P - 2 2 2 1 -1 1 1 det (P) ) - 2 ~ 2 1 - 2 ~1 . A=Ppp's ~ 0 2 20 2 0 0 1 2 -2 1 - 2 O2 -20 1 T . - 2 ㅇ - 0 3 6 02

(b) have the information A= Pop! :- A = (PPP)/pop') (epp) (ppp) (PDF) = PD (PP) 0 (p!p) D (prip) D (P'p) o pri by associative property =PD(I) D(I) P(1) D(I) PP- = PD5 p! ЕР pi 0 0 2 -15 0 -2 (-05 o 25 2 0 0 -2 0 Since diagonal matrix O 0 -2 0 2 0 1 0 32 2 O o TO 1 32 -2 - -62 32 1-2 0 32 -2 -66 0 32

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