5.10.1. Theorem 5.8.

If the rows of a matrix A are dependent, then d( A )=0 .

Proof

Since the rows { A 1 ,, A n }  are dependent, the equation

        i=1 n c i A i =O

can be satisfied with some nonzero c i ’s.  Let c k 0 , then we can write

        A k = ik t i A i               where      t i = c i c k 0

Thus, by the linearity in the kth row, we have

        d( , A k , )=d( , ik t i A i , )   = ik t i    d( , A i , )

Note that every d( , A i , )  in the sum vanishes because the ith and kth rows are identical [see Theorem 5.3(b)].  QED.