Age 16 to 18 Challenge Level:
This question becomes much more meaningful and straightforward if
you consider the geometrical interpretation of the action of a
matrix on a vector.
For example, a rotation matrix will leave a line of points
A matrix which crushes a vector is a type of 'projection'. Clearly
a projection must send a non-zero vector to the origin.
This problem uses mathematical
concepts found in the later Further Pure Mathematics A