Properties of Singular Matrices

[simmonj7]

1. Homework Statement
State whether true or false:
If A and B are singular matrices, then AB is also singular.



3. The Attempt at a Solution
I know that according to the Lemma, if A or B is a singular matrix, then its product AB is also singular. However, it doesn't speak to what happens if two both A and B are singular. I have tried examples using a bunch of singular matrices which I made, and all turned out to be singular. However, I can't get rid of this gut feeling that maybe there is an exception to this situation which I just can't put my finger.

 

[boaz]

In general, if at least one of them is singular then AB is singular, and that's good enough to prove it.
Another way to prove this is using determinant.
Since |A|=0 and |B|=0 (A and B are singular), we get |AB|=|A||B|=0 => AB is singular too.