<s>[INST] problem : conditions of similar matrices Could you please check my solution?
Q. State all conditions that make A and B similar.
$$A=
        \begin{pmatrix}
        0 & 4 \\
        a & 4 \\
      \end{pmatrix}
$$
$$B=
        \begin{pmatrix}
        2 & b \\
        0 & c \\
      \end{pmatrix}
$$
My solution : 
Since similar matrices have the same eigenvalues, trA=4=trB. so C=2. 
DetB=4=DetA so a=-1 
Since both the sum and the product of the two eigenvalues are 4, the eigenvalues of A and B are 2,2 (multiplicity 2)
$$B-2I=
        \begin{pmatrix}
        0 & b \\
        0 & 0 \\
      \end{pmatrix}
$$
$$(B-2I)v=
        \begin{pmatrix}
        0 & b \\
        0 & 0 \\
      \end{pmatrix}\begin{pmatrix}
        x  \\
        y \\
      \end{pmatrix}=\begin{pmatrix}
        by  \\
        0\\
      \end{pmatrix}=\begin{pmatrix}
        0  \\
        0 \\
      \end{pmatrix}
$$, where $$v=\begin{pmatrix}
        x  \\
        y \\
      \end{pmatrix}$$ is an eigenvector of B  
The value of $by$ must always be 0. So b is 0. 
So the answer is a=-1 and b=0 and c=2
 [/INST] Your arguments for $c=2$ and $a=-1$ look convincing, but $b=0$ can't be right. That would lead to $B=2I$, which is not similar to anything except itself.
Your mistake is that you're assuming that the geometric multiplicity of the eigenvalue $2$ is $2$ (such that any $(^x_y)$ would be an eigenvector) whereas this is actually one of the cases where the geometric multiplicity is smaller than the algebraic multiplicity that you've correctly found to be $2$.
Actually $b$ can take any nonzero value, but specifically can't be $0$.
</s>