Question #4138

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.

Expert's answer

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.

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.

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