why use a homogeneous transform matrix?

Pose is described by just 3 numbers:

• Offset in the x-direction
• Offset in the y-direction
• A rotation angle

The homogeneous transformation is inefficient in this respect, requiring three times as many numbers as actually required. Some of those 9 numbers have fixed values like 0 or 1, and others repeated or repeated with a different sign. Its advantage is the ability to compound poses by matrix-matrix multiplication, or transform vectors by matrix-vector multiplication. These operations are very convenient to express in an environment like MATLAB, and with modern computers the extra computation required is almost insignificant.

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn