Can you always find B such that AB=0 when A is Noninvertible?

[buzzmath]

Could anyone help me with this problem?

If A is a noninvertible nxn matrix, can you always find a nonzero nxn matrix B such that AB = 0?

I think you will always be able to find this because a noninvertible matrix can't be reduced to the identity matrix so when row reduced it either has a column or a row that is all zeros. if it's a column that's all zeros say column i then row i in matrix B can be whatever you want and everything else can be zeros. and if row i is all zeros then there will always be a free variable in the matrix to make the product of the two matrices zero. I'm not sure if these are the only cases, if this could really work if the matrix A isn't row reduced, or if this is even a proof or close. Thanks

 

[AKG]

Don't worry about row reducing anything. What can you say about the columns of A? Go from there.