I thought it was a 2 by 1 matrix since they included the addition sign
For a general 2 x 2 matrix, the definition of multiplication is
The *result* of the matrix multiplication is , which is a 2 x 1 matrix, but the actual *operation* (i.e. the thing you do to to get ) is defined by the 2 x 2 matrix with entries a,b,c,d.
[a,0] -> [a, a*sqrt(3)]
[0,b] -> [-b*sqrt(3), b]
Drawing the transformations as triangles, it should be fairly obvious how a general vector is expanded and rotated. The inverse iterations are then rotation in the opposite direction and compression.
It should explain why they ask you to do 6 (inverse) transformations and why the 7th term is powers of 2.