Answer to Question #4138 in Linear Algebra for Kait

Question

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.

Answer 1

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.

2) Let
A =
1 0
0 0

B =
0 0
0 1

Then AB=0.