Question

Question 1 1 pts Cis a 3x3 matrix with exactly two distinct eigenvalues, 11 and 12. Which of the following are possibilities for the algebraic and geometric multiplicities of l, and 12 as eigenvalues of C? (select ALL that apply) It is possible that 11 has algebraic multiplicity 2 and geometric multiplicity 2, and X2 has algebraic multiplicity 1 and geometric multiplicityo. It is possible that X has algebraic multiplicity 2 and geometric multiplicity 1, and 12 has algebraic multiplicity 1 and geometric multiplicity 1. It is possible that X, has algebraic multiplicity 1 and geometric multiplicity 1, and 12 has algebraic multiplicity 1 and geometric multiplicity 2.

Answer 1

SOLUTION O а are Here Hence one of them must for 3x3 matrix three eigen values there. only distinct eigen valus area, triz be repeated I f t repeated algebraic Multiplicit = 2 + Geometric multiplicit or tz is only eigen then algebric multiplis = 1 & Geomem Mulhplzen = ! then 2 then If de repeated Algebraic mulriplicity = 2 4 Seometric mulnplicit lor2 then for ti algebraic mulnplicity / + Scome bic multiplicts=1 Hence correct options possible a FIRST SECOND FOURTH Rest are Not correct as algebraic multiplian con never be less than Geomem Mutriplicit and for last option shows none eigen value repeated SEVENTH

we know A= POPI Here P- Elle [x, X₂ X3 2 o o 5o 1- doo o deo O-4 After calculating pl 2 2 Hence A=PDP' A2 = Pozp' -2 2 [ ES] ] 4 25 o 4 0 16 50 - 16 17 -2 2 21 21 -24 28 12 -18 18 34 A?x -17 21 21 24 28 12 34 [: 25 A²x [ FIRIT OPTION CORRECT