Find number of matrices such that :

Question

The characteristic equation of adjugate of a matrix A is $x^3-x^2+x+1$ , find number of such matrices.

I can't provide my attempt because i don't have any idea on how to handle this question.

Answer 1

The answer depends on the underlying field. You should specify the field when you ask a linear algebra question.

Let $J=\operatorname{adj}(A)$. In general, if $A$ is $n\times n$, then $\det(J)=\det(A)^{n-1}$. If the field is real, from the given characteristic polynomial, we see that $\det(A)^2=\det(J)=-1$, which is impossible because $\det(A)$ is a real number. Hence the answer is zero.

If the field is complex, there are infinitely many possible values of $A$. Let $J$ be the companion matrix of the given characteristic equation. We have found in the previous paragraph that $\det(A)^2=-1$. Suppose $\det(A)=i$. Since $AJ=\det(A)I=iI$, we have $A=iJ^{-1}$. This gives us one solution. But $P^{-1}AP$ is a solution too for every nonsingular matrix $P$. Since $A\ne I$ (otherwise $\det(A)^2=1$, not $-1$), $P^{-1}AP$ assumes infinitely many values when $P$ runs through $GL_3(\mathbb C)$.