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Property of Null Vector watch

1. If a matrix maps all vectors to the null vector, is the determinant of that matrix zero? i.e.

Does

require that ? Can't quite see the solution at the moment and its been awhile since I last looked at this area of maths, so any help would be appreciated.
2. Ax + Dy = 0
Cx + By = 0

Ax = -Dy
Cx = -By

-x/y = D/A = B/C

AB = CD

Something like that.
3. Forgive my ignorance, but if the matrix maps all vectors to the null vector then won't we need A=B=C=D=0 ? i.e. we need the zero matrix, which obviously has a determinant of 0.
4. If the matrix maps any non-zero vector to the null vector then it has determinant 0.
If the matrix maps all vectors to the null vector then it must be the zero matrix.
5. Thank you all for your replies, the misunderstanding highlighted by Kolya and DF is due to my underpar maths speak; I meant any non-zero vector.

Thanks

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Updated: November 11, 2008
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