Show that the product of two reduced row-echelon matrices is also reduced row-echelon.
That's what I think:
- A reduced row-echelon matrix has columns like $e_1 =(1, 0, \cdots , 0)^T$ and $e_2 =(0, 1, 0, \cdots , 0)^T$.
- For columns in between $e_n$ and $e_{n+1}$, only the first $n$ entries will be non-zero.
By noticing these two, I can 'imagine' that the product should be reduced row-echelon. But I cannot write down a clear proof for that. Or say, I don't even know how to start my proof. Can someone give me a helping hand?