Suppose I have three coordinate frames: $A$, $B$ and $C$, all in 2D space. In homogeneous coordinates, I deduce, by inspection, the transformation matrices between each of these ($T_{AB}$, $T_{BC}$ and $T_{AC}$). However, when I calculate $T_{AC}$ analytically, by computing $T_{AC} = T_{AB} * T_{BC}$, I get a different answer to the value of $T_{AC}$ I got by inspection.
Below is an example of this. I'm sure there is something very stupid and simple I am doing wrong, but it's very confusing. Thanks!
Use a version of Gaussian elimination and the idea of linear independence. Then subtracting column 1 from column 3 twice, and adding column 2 to column 3 twice results in your expected output.