1. The problem statement, all variables and given/known data You have a singular 3x3 matrix where det(A-I) = 0 , rank(A+3I) = 2 a) find the characteristic polynomial of A. b) Is A-3I invertible? c) Find det(A) and tr(A) 2. Relevant equations 3. The attempt at a solution a) First I'll find the eigenvalues: - 0 is obviously one since A is singular -det(A-I)=0 => det(I-A) = 0 and so 1 is also an eigenvalue. -rank(A+3I)=2 => rank(-3I - A) = 2 and so det(-3I - A) = 0 and so -3 is also one. So the polynomial is t(t-1)(t+3) b) No because if det(A-3I)=0 => det(3I-A)=0 => 3 is an eigenvalue but we already found 3 and that's the maximum. c) det(A) =0 , and the polynomial is t(t^2+2t -3) = t^3 +2t^2 -3t and so tr(A) = 2. Is that right? Am I missing anything? Thanks.