I was thinking about the following problem:
Let $A$ be a $3\times 3$ matrix with complex entries whose eigenvalues are $1,\pm 2i.$ Suppose that for some $a,b,c\in \mathbb C$, $aA^{-1}=A^{2}+bA+cI,$ where $I$ is the $3\times 3$ identity matrix.Then $(a,b,c)$ equals to which of the following options:
(A)$(-1,-4,4),$
(B)$(4,-1,4),$
(C)$(-1,4,-2),$
(D)$(-1,-2,4).$
My attempts: From $aA^{-1}=A^{2}+bA+cI,$ we get $aI=A^{3}+bA^{2}+cA $ and so trace$(aI)$=trace$(A^{3}+bA^{2}+cA)$=$(\operatorname{trace}(A^{3})+b \operatorname{trace}(A^{2})+ c \operatorname{trace}(A))$.....$(1)$ Now if we take a $3\times 3$ diagonal matrix with diagonal entries $1,\pm 2i $ respectively then,from the above relation $(1)$ we get,$3a=1+b+c$. But from the given options, we see none of them satisfies $3a=1+b+c$.I am sure that i have done some mistake. Please provide a better approach to tackle the problem. Thanks in advance for your time.