What \(3 \times 3\) matrices over \(\mathbb{Q}\) (up to similarity) satisfy \(f(A) = 0\), where \(f(x) = (x^2 + 2)(x - 1)^3\)? List all possible invariant factor lists and rational canonical forms.
Solution
Let \[ f(x)=(x^2+2)(x-1)^3. \] For \(A\in M_3(\mathbb Q)\), the condition \(f(A)=0\) is equivalent to \(\mu_A\mid f\). Since \(x^2+2\) and \(x-1\) are the irreducible factors of \(f\), and \(\deg\chi_A=3\), there are two possible characteristic-polynomial types.
If \[ \chi_A=(x^2+2)(x-1), \] then both coprime irreducible factors must occur in the minimal polynomial, so \[ \mu_A=(x^2+2)(x-1)=\chi_A. \] Thus the invariant-factor list is \[ [(x^2+2)(x-1)], \] and the rational canonical form is \[ C(x^3-x^2+2x-2) =\begin{pmatrix} 0&0&2\\ 1&0&-2\\ 0&1&1 \end{pmatrix}. \]
If \[ \chi_A=(x-1)^3, \] the invariant factors are powers of \(x-1\) whose degrees sum to \(3\) and form a divisibility chain. Hence exactly three lists occur: \[ [x-1,x-1,x-1], \] \[ [x-1,(x-1)^2], \] and \[ [(x-1)^3]. \] Their rational canonical forms are respectively \[ I_3, \] \[ \operatorname{diag}\!\left(1, \begin{pmatrix}0&-1\\1&2\end{pmatrix}\right), \] and \[ \begin{pmatrix} 0&0&1\\ 1&0&-3\\ 0&1&3 \end{pmatrix}. \]
Therefore there are exactly four similarity classes satisfying \(f(A)=0\).