I've been tasked with finding the minimal polynomial of the following matrix: $$A =\left [ \begin{matrix} 0 & 0 & 0 &0 & 0 \\ 1 & 0 & 0 &0 & 0 \\ 0 & 1 & 0 &0 & 0\\ 0 & 0 & 0 &1 & 0 \\ 0 & 0 & 0 &0 & 1 \\ \end{matrix} \right ] $$ Since its triangular, its easy to see that: $p_A = x^3(x-1)^2$
Now, calculating the multiplication of this matrix by itself is a lot of work, and I was wondering if my way of addressing it is correct, and if not what will be the best way to do so.
What I tried doing is dividing the matrix into two blocks as such: $A =\left [ \begin{matrix} B & 0 \\ 0 & C \\\end{matrix} \right ]$ where $B$ is $3\times 3$ and $C$ is $2\times 2$, then finding the Minimal Polynomial for each. which is $x^3$ and and $(x-1)$.
Is that a correct way of doing it? finding the Minimal for each ? because individually, they become zero so does that means the other one can be 'smaller'?
Let $A$ be a diagonal block matrix $\begin{pmatrix} B & 0 \\ 0 & C\end{pmatrix}$.
Then for every polynomial $P$ we have $P(A) = \begin{pmatrix} P(B) & 0 \\ 0 & P(C)\end{pmatrix}$.
It follows $P(A) = I \iff P(B) = I$ and $P(C) = I \iff P$ is a multiple of the minimal polynomial of $B$ and $P$ is a multiple of the minimal polynomial of $C \iff P$ is a multiple of the least common multiple of the minimal polynomials of $B$ and $C$.