RREF of $$$\left[\begin{array}{ccc}1 & 2 & 3\\9 & 12 & 5\\5 & 7 & 4\end{array}\right]$$$

Find the reduced row echelon form of $$$\left[\begin{array}{ccc}1 & 2 & 3\\9 & 12 & 5\\5 & 7 & 4\end{array}\right]$$$.

Subtract row $$$1$$$ multiplied by $$$9$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 9 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & 2 & 3\\0 & -6 & -22\\5 & 7 & 4\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$5$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 5 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & 2 & 3\\0 & -6 & -22\\0 & -3 & -11\end{array}\right]$$$

Divide row $$$2$$$ by $$$-6$$$: $$$R_{2} = - \frac{R_{2}}{6}$$$.

$$$\left[\begin{array}{ccc}1 & 2 & 3\\0 & 1 & \frac{11}{3}\\0 & -3 & -11\end{array}\right]$$$

Subtract row $$$2$$$ multiplied by $$$2$$$ from row $$$1$$$: $$$R_{1} = R_{1} - 2 R_{2}$$$.

$$$\left[\begin{array}{ccc}1 & 0 & - \frac{13}{3}\\0 & 1 & \frac{11}{3}\\0 & -3 & -11\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$3$$$ to row $$$3$$$: $$$R_{3} = R_{3} + 3 R_{2}$$$.

$$$\left[\begin{array}{ccc}1 & 0 & - \frac{13}{3}\\0 & 1 & \frac{11}{3}\\0 & 0 & 0\end{array}\right]$$$

Since the element at row $$$3$$$ and column $$$3$$$ (pivot element) equals $$$0$$$, we need to swap the rows.

Find the first nonzero element in column $$$3$$$ under the pivot entry.

As can be seen, there are no such entries.

The reduced row echelon form is $$$\left[\begin{array}{ccc}1 & 0 & - \frac{13}{3}\\0 & 1 & \frac{11}{3}\\0 & 0 & 0\end{array}\right]\approx \left[\begin{array}{ccc}1 & 0 & -4.333333333333333\\0 & 1 & 3.666666666666667\\0 & 0 & 0\end{array}\right].$$$A