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This non-mathematician needs some help...

Given any 2D transformation matrix as typically used in game programming like this:

$$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ 0 & 0 & 1 \end{pmatrix}$$

Can anyone confirm that such a matrix cannot be used to convert a rectangle into an isosceles trapezoid?

I don't need a proof, I just need to know it's not possible.

I looked at 2D transformation matrix to make a trapezoid out of a rectangle but I couldn't figure out if the answerers were talking about the typical game 2d matrix or other matrices.

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The form of the matrix you quoted corresponds to an affine transformation. One property of these is that they map parallel lines again to parallel lines. So you are right, the two parallel lines of a rectangle (extended beyond the corners to infinite lines) cannot be made to meet, as they would have to for an isosceles trapezoid.

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