Question

(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that tr(A) = 1 and det(A) = 63. Find the eigenvalues of A with their algebraic multiplicities. The smaller eigenvalue has multiplicity and the larger eigenvalue has multiplicity

Answer 1

Soli Ginen That Ä in a 3x3 matre and Ä has only two dustinct eigen values A let A,, a, i, are eigen values of Since Tr (A)=11 det (A) = 63 re Atn 472=1 1, A, A2=63 22 +22=1 27 22=63 Ag = 1-2013 A? (1-2^,)=63 22-223=63 2,2_22, 3-63=0 27-1+6=0 By empection A=-3 is a root -31 2 -1 0 63 lo -6 21 -63 +2 -1 21 to -1 (,+3) (27–29 +21)=0 - *.11=-3 and roots of - 2x²17+21=0 are =-3 non real af , then , ^2 = 1-8(-3)=146=7 .. Eigenvalue of Ä au -3,-3, 7 ... The smallest eigen value =3 has muldpliciti (2] and the largest eigen value = has muldplicity

Sol. Given that, A s 8x matrix A has only two distinct eigen values let ,,A,, a, are eigen values of Ä Ar(A) = 1 and det (A) =63 Atnad 2=1 , 2, 2, 22=63 22,+ 12 = 1 ; 2; 22=63 ^2=1-22, A C1-22,7=63 02-2713=63 27,3+32-63=0 By enspection 2, 3 is a root. 312 1 0 -63 To 6 P 63 + 27 21 le (1-3) (22772+21)=0 11=3,27727+21=0 hos non ralrocks g 2i=3, then D2 =1-6=-5 Eigen values of A are =5,3,3 The smaller eigen value = is has multiplicity I and The larger eigen value = 3 has muldiplicily (2