The question:

A is an c x d matrix. B is a d x k matrix.

If rank(A) = d and AB = 0, show that B = 0.

Solution given:

My textbook has a solution but I don't understand it:

The rank of A is d, therefore A is not the zero matrix. (I asked my prof why d can't be equal to zero, he said it just couldn't...?)

If you left multiply A by some elementary matrix to bring it to row echelon form, you get a matrix that looks like:

[ 1 * * * ... *

0 1 * * ... *

0 0 1 * ... *

0 0 0 0 ... 0] (NOTE: * are arbitrary numbers)

And we will write B as a column (1 x k), consisting of [B1, ... , Bd]T

Multiply A and B together, and you get a column that looks like [R1, R2, ... 0, 0, 0]T

For AB = 0, then Ri = 0. Then since A is not zero, B is 0.

This proof seems to make no sense. Why are we writing B as 1 x k, when the question says B is d x k?

And I don't really get how this actually proves anything... =\