Which homogeneous matrix transforms the figure on the left side

Method to find the correct homogeneous 2D matrix:

• Step 1: Identify three non-collinear points on the original figure and their corresponding images in the transformed figure. A convenient choice visible on the drawings is:

  • Original point (0,2) — the left top corner of the house roof

  • Original point (1,3) — the roof peak

  • Original point (2,2) — the right top corner of the house roof

Step 2: Read off the corresponding target points from the transformed figure.

• The point (0,2) maps to (0,2) (the left apex of the wedge in the transformed figure).

• The point (1,3) maps to (2,3) (the top-left corner where the wedge meets the rectangle).

• The point (2,2) maps to (6,3) (the top-right corner of the rectangle).

Step 3: Set up the affine transform matrix with unknowns and solve.

• Write the homogeneous matrix as [ [a b c], [d e f], [0 0 1] ]. For each chosen point (x,y) -> (x',y') we have x' = ax + by + c and y' = dx + ey + f.

  Using the three correspondences above gives a 6×6 linear system. Solving that system yields:

• Computed affine matrix: [[3, -1, 2], [1/2, 1/2, 1], [0, 0, 1]]

Step 4: Verify the matrix by applying it to the three original points:

• (0,2) -> x' = 3*0 + (−1)*2 + 2 = 0 ; y' = 0.5*0 + 0.5*2 + 1 = 2 ⇒ (0,2)

• (1,3) -> x' = 3*1 + (−1)*3 + 2 = 2 ; y' = 0.5*1 + 0.5*3 + 1 = 3 ⇒ (2,3)

• (2,2) -> x' = 3*2 + (−1)*2 + 2 = 6 ; y' = 0.5*2 + 0.5*2 + 1 = 3 ⇒ (6,3)

Conclusion:

• The correct affine transform (in homogeneous coordinates) that matches the picture is [[3, -1, 2], [1/2, 1/2, 1], [0, 0, 1]].

• None of the four provided matrices equals this matrix, so none of the supplied answer choices is correct as written.

• If the intention was to use integer entries only, check whether a different selection of sample points or a transcription error in the choices occurred; but based on the drawing and the standard method above, the computed matrix is the correct affine transform.