A 3×33 \times 33×3 real matrix AAA is never shown to you. All you are told is
tr(A)=6,tr(A2)=44,det(A)=−24\operatorname{tr}(A) = 6, \qquad \operatorname{tr}(A^2) = 44, \qquad \det(A) = -24tr(A)=6,tr(A2)=44,det(A)=−24
and that all three eigenvalues of AAA are real.
Find the eigenvalues of AAA, and then evaluate det(A−3I)\det(A - 3I)det(A−3I).
Select all that apply.
Eigenvalues {6, 2, −2}\{6,\ 2,\ -2\}{6, 2, −2} and det(A−3I)=−15\det(A - 3I) = -15det(A−3I)=−15
Eigenvalues {8, 1, −3}\{8,\ 1,\ -3\}{8, 1, −3} and det(A−3I)=60\det(A - 3I) = 60det(A−3I)=60
Eigenvalues {6, 2, −2}\{6,\ 2,\ -2\}{6, 2, −2} and det(A−3I)=−51\det(A - 3I) = -51det(A−3I)=−51
Eigenvalues {6, 2, −2}\{6,\ 2,\ -2\}{6, 2, −2} and det(A−3I)=15\det(A - 3I) = 15det(A−3I)=15