Order of affine transformations on matrix

Question

I am trying to solve the following question:

Apparently the correct answer to the question is (a) but I can't seem to figure out why that is the case.

The only way I can seem to replicate the matrix is to first do a translation by (-2,2) and then rotating by 90 degrees. However, the answer says that:

M represents a translation of vector (2,2) followed by a rotation of angle 90 degrees transform

If it is a translation of (2,2), then why does the matrix M not contain (2,2,1) in its last column?

Answer 1

Allow me to do the calculation in a computer algebra system (Octave):

First the translation:

>> T = [ 1,0,2; 0,1,2; 0,0,1]
T =

   1   0   2
   0   1   2
   0   0   1

Then the rotation by $90^\circ$:

>> R = [ 0, -1, 0; 1, 0, 0; 0,0,1]
R =

   0  -1   0
   1   0   0
   0   0   1

Finally the composition of the two, applying first $T$ then $R$:

>> R*T
ans =

   0  -1  -2
   1   0   2
   0   0   1

So this is a consequence of performing the matrix multiplication with this specific $R$.