row reduced augmented matrix

[Sonarks]

Thought i knew how to use Gaussian elimination to find the row echelon form but i just cant seem to get the answer the attached question is looking for in part b. Can someone pleaseeeeeee help

 

[MarkFL]

Hello, and welcome to FMH!

 

[Sonarks]

Hello, and welcome to FMH!

(a) [MATH]\left[\begin{array}{c}1 & 2 & -1 & -3 \\ 3 & 5 & k & -4 \\ 9 & k+13 & 6 & 9 \end{array}\right][/MATH]
(b) \(-3R_1+R_2\)

[MATH]\left[\begin{array}{c}1 & 2 & -1 & -3 \\ 0 & -1 & k+3 & 5 \\ 9 & k+13 & 6 & 9 \end{array}\right][/MATH]
\(-R_2\)

[MATH]\left[\begin{array}{c}1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 9 & k+13 & 6 & 9 \end{array}\right][/MATH]
\(-9R_1+R_3\)

[MATH]\left[\begin{array}{c}1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 0 & k-5 & 15 & 36 \end{array}\right][/MATH]
\(-(k-5)R_2+R_3\)

[MATH]\left[\begin{array}{c}1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 0 & 0 & k^2-2k & 5k+11 \end{array}\right][/MATH]

Thank you so much