Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$
Show that the characteristic and minimal polynomials of $A$ are the same.
I have already computated the characteristic polynomial
$$p_A(x)=x^3-ax^2-bx-c$$
and I know from here that if I could show that the eigenspaces of $A$ all have dimension $1$, I would be done. The problem is that solving for the eigenvalues of this (very general) cubic equation is difficult (albeit possible), meaning it would be difficult to find bases for the eigenspaces.
A hint would be appreciated.
Best Answer
Compute: $$A^2 = \begin{pmatrix} 0 & c & ac \\ 0 & b & c + ab \\ 1 & a & b + a^2\end{pmatrix}.$$ So, we just need to show that $A^2, A, I$ are linearly independent. Clearly $A$ is not a multiple of $I$, so we just need to show there is no solution to the equation $$A^2 = pA + qI \iff \begin{pmatrix} 0 & c & ac \\ 0 & b & c + ab \\ 1 & a & b + a^2\end{pmatrix} = p\begin{pmatrix} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a\end{pmatrix} + q\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$ for $p$ and $q$. In particular, if you examine the entries in the left column, bottom row, we get $$1 = 0p + 0q,$$ which means there is indeed no solution. Hence $I, A, A^2$ are linearly independent, so no quadratic of $A$ will be equal to the $0$ matrix. Thus, the minimal polynomial must be (at least) a cubic, and equal to the characteristic polynomial