Simple linear algebra justification - clarification

• Feb 3rd 2010, 01:06 PM
gate13
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.

• Feb 3rd 2010, 03:11 PM
tonio
Quote:

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.

Again, looks fine to me.

Tonio
• Feb 3rd 2010, 03:38 PM
gate13
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.
• Feb 3rd 2010, 06:56 PM
math2009
ref picture
• Feb 6th 2010, 09:00 PM
gate13
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