Fixed point of the plane isometry
Fixed points coincide with their image X=X'.
Any direct isometry could be viewed as composition of rotation about origin and translation. The 3rd column of matrix representation A is image of origin, not center of rotation. Center is fixed point of isometry.
Experiment with arbitrary point X and its image X'. Could you write the equation for fixed points of the mapping?
Fixed points
It is easy to derive equation for fixed points:
Gauss elimination (GeoGebra command 
Using back-substitution, unknown coordinates x, y can be solved for.
Transformation has only one fixed point FP = (2.5, 3.5).
Gauss elimination (GeoGebra command ReducedRowEchelonForm) of the matrix (A-E) returns the equivalent echelon form
Using back-substitution, unknown coordinates x, y can be solved for.
Transformation has only one fixed point FP = (2.5, 3.5).