I have the upper triangular matrix
$A = \begin{bmatrix} 1 & 32 & \ast & \dots & \ast\\ 0 & 1 & \ast & \dots & \ast\\ 0 & 0 & 21 & \dots & \ast \\ \vdots & \vdots & \vdots & \ddots & \ast\\ 0 & 0 & 0 & \dots & \ast \\ \end{bmatrix}$
where $\ast$ represents any number.
I know $A$ is not diagonalisable, and want to prove $(x-1)^2$ divides the minimum polynomial, $m_A(x)$, for $A$.
I know $A$ is diagonalisable if and only if $m_A(x)$ is a product of distinct linear factors. Hence $m_A(x)$ is not the product of distinct linear factors.
I also know that $m_A(x)$ divides the characteristic polynomial of $A$. Hence $m_A(x)$ divides $(x-1)^2f(x)$.
Are these two facts enough to prove that $(x-1)^2$ must divide $m_A(x)$? I am unsure whether $f(x)$ may contain the repeated linear factors?