Prove that $A$ is similar to $B$ probably using Jordan form

Question

Let $A, B \in M_n(\mathbb{F})$ such that:

1. $a_{ij} = 0 \iff b_{ij} = 0$.
2. $a_{ij} = b_{ij}$ for all $i \ne j+1$
3. $\exists \lambda \in \mathbb{F}$ such that $a_{ii} = b_{ii} = \lambda$.

Prove that $A$ and $B$ are similar matrices.

This question should be related (I think) to the Jordan form material. I'd be glad for any help.

Thanks.

Answer 1

I think you've mis-stated some of your conditions. As I understand them, the following two matrices satisfy your three conditions but they are not similar: $$\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} \qquad \text{and} \qquad \begin{bmatrix} 2 & 2 \\ 1 & 2 \end{bmatrix}$$