Thread: Simple linear algebra justification - clarification

Good day to all.

I have been reading on transposes and came across the following question:

If A is a n x n matrix where A is non zero can AA^T = 0

I thought about it and came to the conclusion that the answer is no with the following justification:

Since A is a non zero matrix then at the very least some (A)i,j element is <> 0.

By definition of the transpose this implies that the (A)j,i element is <> 0 and (A)j,i = (A)i,j

Therefore the cross product of the ith row of A with the ith column of A^T will have a minimum value of ((A)i,j)^2

Does this conclusion make sense or have I missed something along the way.

Thanks for your help

Originally Posted by gate13

Good day to all.

I have been reading on transposes and came across the following question:

If A is a n x n matrix where A is non zero can AA^T = 0

I thought about it and came to the conclusion that the answer is no with the following justification:

Since A is a non zero matrix then at the very least some (A)i,j element is <> 0.

By definition of the transpose this implies that the (A)j,i element is <> 0 and (A)j,i = (A)i,j

Therefore the cross product of the ith row of A with the ith column of A^T will have a minimum value of ((A)i,j)^2

Does this conclusion make sense or have I missed something along the way.

Thanks for your help

Again, looks fine to me.

Tonio

Thanks tonio for your input. I have become somewhat paranoid with my linear algebra TA as he tends to not be very accepting of answers that deviate slightly from his. At least this way, I know that I have not misunderstood the material and I can attempt to vigorously defend my point of view. Again many thanks for your input.

ref picture

Thanks math2009for the attached document. It will be very useful, although right now we have not covered images yet. That portion of the proof I will look into on my own so that I may understand it. Finally I apologize for my tardy reply. I have been wrapped up in mid-term studies. Again many thanks