Let A be the 4x4 matrix with elements in positions (1,2), (2,3), (3,4), and (4,1) equal to 1, and all other elements equal to zero. Let b = $\displaystyle e_1 and x_0$=0. Show that $\displaystyle ||r_k||_2=||r_0||_2$ for k=1,2,3.

Here, $\displaystyle r_k=Ax_k-b$ where $\displaystyle x_k$ is the iterate produced by GMRES in iteration k.

Finding $\displaystyle r_0$ is not a problem for me since $\displaystyle x_0$ is the zero vector. Therefore, $\displaystyle ||r_0||$= b. However, we have not learned how to actually figure out the different iterations, so I'm guessing that we do not have to find $\displaystyle x_1$,$\displaystyle x_2$, and $\displaystyle x_3$, explicitly, but I have no idea how to show that $\displaystyle ||r_k||_2$=b for k=1,2,3. Can anyone push me in the right direction? Thank you.