Answer to Question #4138 in Linear Algebra for Kait
Suppose A is an m x n matrix and B is an n x l matrix. Further, suppose that A has a row of zeros. Does AB have a row of zeros? Why or why not? Does this also hold true if B has a row of zeros? Why or why not?
Give an example of two matrices A and B for which AB=0 with A not equal to 0 and B not equal to 0.
1) Regard rows of A as n-vectors and columns of B Notice that each row of A and each column of B are n-vectors. Then (i,j)-element of AB is the dot product of i-th row of A and j-th column of B. So if A has a i-th row of zeros, then for each j=1,...,l the (i,j)-element of AB is zero, so AB iwll also have i-th row of zeros.
The assumption that B has a row of zeros, does not imply that AB has a row of zeros, but by the same arguments as above, j-th column of AB will consist of zeros.
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