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.
$A =
\begin{bmatrix}
    1&3&4&5\\3&9&12&9\\1&3&4&1
\end{bmatrix}$
$B=
\begin{bmatrix}
    1&2&1&2\\0&1&0&1\\-1&2&0&3
\end{bmatrix}$
 A: 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.
A: Reduced row echelon form is a specific echelon form of a matrix that can be obtained by modifying a matrix with basic row operations, such that every leading coefficient is 1 and is the only nonzero entry in its column.
Useful: http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=rref
A: You begin to put the matrix in echelon form, with the pivots equal to $1$, going downwards row after row. When that is done, all coefficients under a pivot (in the same column) are equal to $0$.Then you make the coefficients above the pivots equal to $0$, starting fom the last pivot and going upwards:
\begin{align*}
&\begin{bmatrix}
1&3&4&5\\3&9&12&9\\1&3&4&1
\end{bmatrix}
\rightsquigarrow
\begin{bmatrix}
1&3&4&5\\0&0&0&-6\\0&0&0&-4\\
\end{bmatrix}
\rightsquigarrow
\begin{bmatrix}
1&3&4&5\\0&0&0&1\\0&0&0&1
\end{bmatrix}\\
\rightsquigarrow
&\begin{bmatrix}
1&3&4&5\\0&0&0&1\\0&0&0&0
\end{bmatrix}\rightsquigarrow
\begin{bmatrix}
1&3&4&0\\0&0&0&1\\0&0&0&0
\end{bmatrix}
\end{align*}
