Let $A\in M_n(\Bbb{C})$ such that $\text{Im}(A)\cap\text{Ker}(A)=\theta$

Question

The actual question looks like-
Let $A\in M_n(\Bbb{C})$ such that $\text{Im}(A)\cap \text{Ker}(A)=\{\theta\}$, where $\text{Im}(A)=\{AX \mid X\in\Bbb{C^n}\}$ and $\text{Ker}(A)=\{X\in\Bbb{C}^n\mid AX=\theta$}, then prove that there exists non-singular matrices $P$ and $D$ of orders $n\times n$ and $\text{rank}(A)\times\text{rank}(A)$ respectively such that $$ A=P \begin{pmatrix} D & 0 \\ 0 & 0 \\ \end{pmatrix} P^{-1} $$ Can anybody suggest me a proper solution to that question?
Thanks for assistance in advance.

Answer 1

You can consider a basis of $\mathbb{C}^n$ using the decomposition $\mathbb{C}^n = Im(A) \oplus Ker(A)$.