I think you made a few typos. Det should be Let, and x(c) should be x(0).
The matrix A transforms a vector by rotating it counterclockwise 45 degrees.
Recall the general form for rotation matrices
[cos(Theta) -sin(Theta) ]
[sin(Theta) cos(Theta) ]
using Theta = 45 degrees gives the matrix A
So x(1) is x(0) rotated by 45 degrees. x(2) is x(1) rotated by 45 degrees and so on...