some confusion about minial polynomial.?

Question

let $A = \begin{bmatrix} c_1 & 0\\ 0 &c_2 \end{bmatrix}$

$B=\begin{bmatrix}c_1&0\\1&c_2\end{bmatrix}$

I thinks Both A and B have same minimial polynomial because both have same characteristics polynomials...

Answer 1

If two $n{\,\times\,}n$ matrices have the same minimal poynomial, they have the same characteristic polynomial.

But the converse need not hold.

However, for an $n{\,\times\,}n$ matrix, the minimal and characteristic polynomials have the same set of distinct eigenvalues, hence in this case, if $c_1\ne c_2$, the polynomial $(x-c_1)(x-c_2)$ is the minimal and characteristic polynomial for both $A$ and $B$.

On the other hand, if $c_1=c_2 = c$, then $A$ and $B$ share the characteristic polynomial $(x-c)^2$, but the minimal polynomial for $A$ is $x-c$, while the minimal polynomial for $B$ is $(x-c)^2$.