Html5 Canvas Transformation Algorithm - Finding object coordinates after applying transformation

Question

On html5 canvas I am drawing objects (rectangle, circle, etc...), these objects have transformation properties like scale, skew, rotation etc... These objects can be nested.

Answer 1

Tom Larkworthy's answer is perfect, but with a minor typo. Correct formula to invert the matrix is:

[  d/M  -c/M  (c*dy - d*dx)/M ]
[ -b/M   a/M  (b*dx - a*dy)/M ]
[   0     0          1        ]