Graphical transformations

STEP 1998, Math III, #5: The exponential of a square matrix

is defined to be

where and is the identity matrix.

Let

and hence express as a single 2X2 matrix, where is a real number. Explain the geometrical significance of .

Let

Express similarly , where is a real number, and explain the geometrical significance of .

For which values of

does

for all

? Interpret this fact geometrically.