does $\displaystyle A=\left(\begin{array}{cc}a & 1\\0 & a\end{array}\right)$ and

$\displaystyle C=\left(\begin{array}{cc}a & b\\ 0 & a\end{array}\right)$

are similar?

i know a law that if they have the same jordan form then they are similar

i was told that if b=0 they the dont have the same jordan form

if b=0

jordan form of A:

$\displaystyle P(t)=(a-t)^{2}$

if the minimal polinomial is

M(t)=(a-t) then $\displaystyle M(A)=(aI-A)\neq0$

so $\displaystyle M(t)=(a-t)^{2}$

so $\displaystyle j_{A}=\left(\begin{array}{cc}a & 1\\0 & a\end{array}\right)$same form i will get for C

where is my mistake?