Property of Null Vector Watch

div curl F = 0
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#1
Report Thread starter 10 years ago
#1
If a matrix maps all vectors to the null vector, is the determinant of that matrix zero? i.e.

Does
 \left( \begin {array}{cc}A&D\\\noalign{\medski  p}C&B\end {array} \right) \left( \begin {array}{c}x\\\noalign{\medskip}y  \end {array} \right) = \left( \begin {array}{c}0\\\noalign{\medskip}0  \end {array} \right)

require that  AB=CD? 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.
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Juwel
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#2
Report 10 years ago
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Ax + Dy = 0
Cx + By = 0

Ax = -Dy
Cx = -By

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

AB = CD

Something like that.
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Kolya
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#3
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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.
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DFranklin
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#4
Report 10 years ago
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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.
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div curl F = 0
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#5
Report Thread starter 10 years ago
#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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