a question about substitution Matrix in MWG book

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there is a  proposition in the 35th page  in the textbook of Micoeconomic theory written by MWG as following
Proposition 2.F.3 Suppose that the walrasion demand function x(p,w) is differentiable,homogeneous of degree zero, and satisfies Walras's law. Then
, the author said that "It follows from Proposition 2.F.3 that the martrix S(p,w) is always singlular(i.e., it has rank less than L)"

then we can conclude that  "pS(p,w)=0 and S(p,w)p=0 for any (p,w)" implies that S(p,w) is singular.  How to understand the conclusion?
I  know that  the necessary and sufficient condition for S(p,w) being singular is |S(p,w)|=0.

I'm waiting for your help.  


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关键词：substitution question matrix About Quest matrix MWG Book substitution question

I think you can understand this conclusion using the definition of the rank of a matrix. pS(p,w) is a linear combination of the rows of the matrix S(p,w).
From pS(p,w)=0 and p is not zero, you can see that the row vectors of S(p,w) are linear dependent, which means the rank of S(p,w) must be less than L.

|p|*|S(p,w)| = |0| = |S(p,w)|*|p| = 0

|p| = 0不成立，所以|S(p,w)| = 0

|p| 是什么？

xuqiuhua 发表于 2011-5-12 19:08
|p| 是什么？

p的行列式啊。
所以sp=0且p=/=0说明s是线性相关的。所以s不满秩。当然反过来推也成立。所以是充要条件