[Reduced] Column Echelon Form

I'm trying to understand this a little better.

If we have a $\displaystyle n\times n$ matrix, $\displaystyle \left[\begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1n}\\a_ {21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots& a_{nn}\end{array}\right]$, would the CEF have the form $\displaystyle \left[\begin{array}{ccccc}1&0&0&\cdots&0\\a_{21}&1&0&\cd ots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\a_{(n-1)1}&a_{(n-1)2}&\cdots&1&0\\a_{n1}&a_{n2}&a_{n3}&\cdots&1\end {array}\right]$??

And would the RCEF of this matrix be $\displaystyle I_{n\times n}$??\

I'd appreciate any clarification!

--Chris