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...