<s>[INST] what is the characteristic polynomial and minimal polynomial of A and B? What is the characteristic polynomial  and minimal polynomial  of $A$  and $B$ ?
$A=\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0  \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix},B=\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0  \\ 0 & 0 & a & 1 \\ 0 & 0 & 0 & a\end{pmatrix}$
My attempt :  for  $A$)  , $ch_A(x) = (x-a)^4$ , $m_A = (x-a)^2(x-a)$
for $B)$ $ch_B(x) = (x-a)^4$ , $m_B = (x-a)^2(x-a)^2$
where $ch$ and $m$ denote  the characteristic and minimal polynomial.
Is  my answer is correct or not  ???
Any hints/solution will be apprciated
thanks u
 [/INST] $A=\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0  \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix},$  first  i break them into  two jordan nlock forms  that $P=\begin{pmatrix} a & 1   \\ 0 & a \end{pmatrix}$ and $Q=\begin{pmatrix} a & 0   \\ 0 & a \end{pmatrix}$
now , the characteristic and minimal polynomial of $P = (\lambda -a)^2=f(t)$
and the characteristic and minimal polynomial of $Q = (\lambda -a)^2=g(t)$
Now the minimial polynomial of A  $m_A = Lcm(f(t),g(t))=lcm\{(\lambda -a)^2,(\lambda -a)^2\}=(\lambda -a)^2$
similarly for  B matrix  we get $m_B = Lcm(f'(t),g'(t))=lcm\{(\lambda -a)^2,(\lambda -a)^2\}=(\lambda -a)^2$
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