Linear algebra Upper Triangular matrix

I have a question regarding upper triangular matrix.

I know that if AB is upper triangular then the |AB| equals to the diagnoal multiplication, but It doesn't seem to help me here.

If AB is upper triangular and non-singular then A and B both upper triangular?

Notice that A and B are both square.

I can't choose AB = 0 becuase its signular....

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Are $A,B$ square? – wj32 Oct 29 '12 at 8:35
Yes they are both square matrices – SyndicatorBBB Oct 29 '12 at 8:36
Never mind, didn't read the question properly... – wj32 Oct 29 '12 at 8:37

No, $A$ and $B$ does not necessarily have to be upper triangle. For example, $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$
I was wondering about one adjugate matrix property. I know that If $A$ is symmetric so is $adj(A)$. I was thinking to myself, is the opposite direction is also TRUE? If $adj(A)$ is symmetric then $A$ is symmetric ? Thank you. – SyndicatorBBB Oct 29 '12 at 8:49
@Guy: Take a $3 \times 3$ matrix with a 1 in the upper-right corner (or anywhere). – wj32 Oct 29 '12 at 8:54