RREF of $$$\left[\begin{array}{ccc}11 & -2 & 8\\8 & 1 & 3\\-2 & -7 & 7\end{array}\right]$$$

Find the reduced row echelon form of $$$\left[\begin{array}{ccc}11 & -2 & 8\\8 & 1 & 3\\-2 & -7 & 7\end{array}\right]$$$.

Divide row $$$1$$$ by $$$11$$$: $$$R_{1} = \frac{R_{1}}{11}$$$.

$$$\left[\begin{array}{ccc}1 & - \frac{2}{11} & \frac{8}{11}\\8 & 1 & 3\\-2 & -7 & 7\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$8$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 8 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & - \frac{2}{11} & \frac{8}{11}\\0 & \frac{27}{11} & - \frac{31}{11}\\-2 & -7 & 7\end{array}\right]$$$

Add row $$$1$$$ multiplied by $$$2$$$ to row $$$3$$$: $$$R_{3} = R_{3} + 2 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & - \frac{2}{11} & \frac{8}{11}\\0 & \frac{27}{11} & - \frac{31}{11}\\0 & - \frac{81}{11} & \frac{93}{11}\end{array}\right]$$$

Multiply row $$$2$$$ by $$$\frac{11}{27}$$$: $$$R_{2} = \frac{11 R_{2}}{27}$$$.

$$$\left[\begin{array}{ccc}1 & - \frac{2}{11} & \frac{8}{11}\\0 & 1 & - \frac{31}{27}\\0 & - \frac{81}{11} & \frac{93}{11}\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$\frac{2}{11}$$$ to row $$$1$$$: $$$R_{1} = R_{1} + \frac{2 R_{2}}{11}$$$.

$$$\left[\begin{array}{ccc}1 & 0 & \frac{14}{27}\\0 & 1 & - \frac{31}{27}\\0 & - \frac{81}{11} & \frac{93}{11}\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$\frac{81}{11}$$$ to row $$$3$$$: $$$R_{3} = R_{3} + \frac{81 R_{2}}{11}$$$.

$$$\left[\begin{array}{ccc}1 & 0 & \frac{14}{27}\\0 & 1 & - \frac{31}{27}\\0 & 0 & 0\end{array}\right]$$$

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 $$$\left[\begin{array}{ccc}1 & 0 & \frac{14}{27}\\0 & 1 & - \frac{31}{27}\\0 & 0 & 0\end{array}\right]\approx \left[\begin{array}{ccc}1 & 0 & 0.518518518518519\\0 & 1 & -1.148148148148148\\0 & 0 & 0\end{array}\right].$$$A