1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. The attempt at a solution 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? It says in the question B is d x k! Also if A is not zero then why can't you say right off the bat that AB = 0 implies B =0?