linear algebra - Prove that every $A in M_nleft ( mathbb{C} right )$ is similar to a matrix with at most one non-zero element in the first column

I need that prove that every $A \in M_n\left ( \mathbb{C} \right )$ is similar to a matrix $B$ where $B$'s first column is of the form $\begin{pmatrix}\lambda\\0\\\vdots\\ 0\\ \end{pmatrix}$

where $M_n\left ( \mathbb{C} \right )$ is the set of all square matrices above $\mathbb{C}$.

I haven't been able to make much progress with this question - any help would be appreciated.

Answer

Every complex matrix is triangularizable, because its characteristic polynomial factorises completely into linear factors. Hence A is similar to an upper-triangular matrix. The first column of such a matrix has the desired form.