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Fix Me or Crush Me

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 invariant.

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 Level modules.