Question

2. (6 pts) The matrix 4 2 -15 -1 2 1 4 has spectrum o(A) = {3,5}. (a) Determine the geometric multiplicity of each eigenvalue. (b) Is either eigenvalue deficient? If so, which? If not, how can you tell?

Answer 1

Que 2. (a). 4 2 -L Solution. Consider , Consider. As -1 5 I .L 2 4 So are th and A2 = 3 12 = 5 distinct eigen of A. Since spectrum o(A) = 43,5} the only two values Now, Greometric multiplicity (6.M.) of eigen valere A of matrix Nullity of [A-11] m (A-AI) ^ (^~31) — 0 A G.M. of 1,=3 of (A-31] L 2 I 11 - 2 I 2 1 1 2 (R3 = R3-R2) N -1 2 -) m (A-3 I) EL . GoM. of do= 3 of A FI Now , [A-E -51] Oo 2 L r - O r 1 | 2 -1 r 2 - (R'S = R3-R) -L 0 0 m (A-5I) = 1

:: Gim. of 12=5 of A = 1 (6). Now, consider IA-I = 0 - -1 -) 4-4 -1 2 5-) I -1 2 44 co >> (4-1) ((5-1) (4-2) -2 ) -2[-(4-3) +1] -1 [-2 +15–1] =0 -) (5-0) (4-1)2-214-2) + 2 (470) -2 +2 – (5-1) =0 =) (5-4) [4-1)2 - 1) 3) (5-0) [(5-1) (3-0) ] = =) ie algebraic multiplicity of dq = 3 in '' and algebraic multiplicity (A.m.) of de 55 in '2' ( 3,5,5 1 ie A.M. A1 = 3 A of of of A.M. ox A 15 2 of A2=5 of A. but of 12=5 of A. from equation ☺ ☺ ☺ 8 5 A.M. of di = 3 of G.M. of ^, =3 of A A.M. of A2 = 5 of A & G.M. Sime any eigen value is called deficient it it's A.M. is not equal to G.M. and not deficient if it's A.M. is equal to GM. Therefore A = 5 is deficient eigen value of while Nr = 3 is not deficient eigen is not deficient eigen value of A. A