Endomorphism with a parameter

Let $f_k$ be the endomorphism of ${\mathbb{R}}^3$ with matrix

$$A_k=\left(\begin{array}{ccc}k& 1& 1\\ 1& k& 1\\ 0& 1& 1\end{array}\right).$$

Find the kernel and the image as $k\in \mathbb{R}$ varies.