Let $A \in M_n(\mathbb R)$ and suppose its minimal polynomial is: $M_A(t)=\prod_{i=1}^{k}(t-\lambda_i)^{\textstyle s_i}.$
When $\lambda _1,\lambda_2,\lambda _3,......,\lambda _k$ are distinct eigenvalues.
We define a new matrix: $B\in M_{2n}(\mathbb R)$ by: $\left(\begin{matrix} A &I_n \\ 0 & A \end{matrix}\right)$
I need to prove that the minimal polynomial of $B$ is $M_B(t)=\prod_{i=1}^k(t-\lambda_i)^{\textstyle s_{i}+1}.$
Edit: What follows below refered to the original version of this question, in which the definition of $B$ was: $B = \left(\begin{array}{cc}A&I_n\\I_n&A\end{array}\right).$
I tried to do that in induction. I don't understand this basic case: for $1$x$1$ matrix: $\begin{pmatrix} 5 \end{pmatrix}$ we get that $ \begin{pmatrix} 5 & 1\\ 1& 5 \end{pmatrix}$ 's minimal polynomial is $(x-4)(x-6)$ and it doesn't answer the condition. so I have also a problem with understanding the question I guess.
Thanks!!