What is row reduced echelon form? How to row reduce this matrix? I'm not being able to grasp the concept of row reduced echelon form. Please, explain how to row reduce one of the the following matrices.
$A =
\begin{bmatrix}
    1&3&4&5\\3&9&12&9\\1&3&4&1
\end{bmatrix}$
$B=
\begin{bmatrix}
    1&2&1&2\\0&1&0&1\\-1&2&0&3
\end{bmatrix}$

In order to obtain the reduced row echelon form (rref) of a matrix, we apply some row operations. According to this article, for the first case we have:
$\begin{array}{l} \begin{bmatrix} 1 & 3 & 4 & 5\\ 3 &  9 & 12 & 9\\ 1 & 3 & 4 & 1 \end{bmatrix}\overset{R_2:=3R_1 - R_2}{\to}\begin{bmatrix} 1 & 3 & 4&5\\0&0&0&6 \\ 1&3&4&1 \end{bmatrix}\overset{R_3:=R_1 - R_3}{\to}\begin{bmatrix} 1&3&4&5\\0&0&0&6\\0&0&0&4 \end{bmatrix}\overset{R_2:=\frac 16R_2}{\to}\begin{bmatrix}1&3&4&5\\ 0&0&0&1\\0&0&0&4\\\end{bmatrix}\\\overset{R_3:=4R_2-R_3}{\to}
\begin{bmatrix}1&3&4&5\\0&0&0&1\\0&0&0&0\end{bmatrix}\overset{R_1:=R_1 - 5R_2}{\to}\begin{bmatrix}1&3&4&0\\ 0&0&0&1\\0&0&0&0 \end{bmatrix}.
\end{array}$
The last matrix satisfies all the conditions of the reduced row echelon form of a matrix.
