RREF of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & a & -1\\2 & a & 2 & 4 & a\\-1 & -2 & - a & 1 & 1\end{array}\right]$$$

Find the reduced row echelon form of $$$\left[\begin{array}{ccccc}1 & 3 & -3 & a & -1\\2 & a & 2 & 4 & a\\-1 & -2 & - a & 1 & 1\end{array}\right]$$$.

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & a & -1\\0 & a - 6 & 8 & 4 - 2 a & a + 2\\-1 & -2 & - a & 1 & 1\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & a & -1\\0 & a - 6 & 8 & 4 - 2 a & a + 2\\0 & 1 & - a - 3 & a + 1 & 0\end{array}\right]$$$

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

$$$\left[\begin{array}{ccccc}1 & 3 & -3 & a & -1\\0 & 1 & \frac{8}{a - 6} & \frac{4 - 2 a}{a - 6} & \frac{a + 2}{a - 6}\\0 & 1 & - a - 3 & a + 1 & 0\end{array}\right]$$$

Subtract row $$$2$$$ multiplied by $$$3$$$ from row $$$1$$$: $$$R_{1} = R_{1} - 3 R_{2}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & - \frac{3 \left(a + 2\right)}{a - 6} & \frac{a^{2} - 12}{a - 6} & - \frac{4 a}{a - 6}\\0 & 1 & \frac{8}{a - 6} & \frac{4 - 2 a}{a - 6} & \frac{a + 2}{a - 6}\\0 & 1 & - a - 3 & a + 1 & 0\end{array}\right]$$$

Subtract row $$$2$$$ from row $$$3$$$: $$$R_{3} = R_{3} - R_{2}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & - \frac{3 \left(a + 2\right)}{a - 6} & \frac{a^{2} - 12}{a - 6} & - \frac{4 a}{a - 6}\\0 & 1 & \frac{8}{a - 6} & \frac{4 - 2 a}{a - 6} & \frac{a + 2}{a - 6}\\0 & 0 & - \frac{\left(a - 5\right) \left(a + 2\right)}{a - 6} & \frac{\left(a - 5\right) \left(a + 2\right)}{a - 6} & - \frac{a + 2}{a - 6}\end{array}\right]$$$

Multiply row $$$3$$$ by $$$- \frac{a - 6}{\left(a - 5\right) \left(a + 2\right)}$$$: $$$R_{3} = - \frac{a - 6}{\left(a - 5\right) \left(a + 2\right)} R_{3}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & - \frac{3 \left(a + 2\right)}{a - 6} & \frac{a^{2} - 12}{a - 6} & - \frac{4 a}{a - 6}\\0 & 1 & \frac{8}{a - 6} & \frac{4 - 2 a}{a - 6} & \frac{a + 2}{a - 6}\\0 & 0 & 1 & -1 & \frac{1}{a - 5}\end{array}\right]$$$

Add row $$$3$$$ multiplied by $$$\frac{3 \left(a + 2\right)}{a - 6}$$$ to row $$$1$$$: $$$R_{1} = R_{1} + \frac{3 \left(a + 2\right)}{a - 6} R_{3}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 0 & a + 3 & - \frac{4 a + 1}{a - 5}\\0 & 1 & \frac{8}{a - 6} & \frac{4 - 2 a}{a - 6} & \frac{a + 2}{a - 6}\\0 & 0 & 1 & -1 & \frac{1}{a - 5}\end{array}\right]$$$

Subtract row $$$3$$$ multiplied by $$$\frac{8}{a - 6}$$$ from row $$$2$$$: $$$R_{2} = R_{2} - \frac{8}{a - 6} R_{3}$$$.

$$$\left[\begin{array}{ccccc}1 & 0 & 0 & a + 3 & - \frac{4 a + 1}{a - 5}\\0 & 1 & 0 & -2 & \frac{a + 3}{a - 5}\\0 & 0 & 1 & -1 & \frac{1}{a - 5}\end{array}\right]$$$

The reduced row echelon form is $$$\left[\begin{array}{ccccc}1 & 0 & 0 & a + 3 & - \frac{4 a + 1}{a - 5}\\0 & 1 & 0 & -2 & \frac{a + 3}{a - 5}\\0 & 0 & 1 & -1 & \frac{1}{a - 5}\end{array}\right]$$$A.