I am completing a practice questions sheet for the topic "systems of linear of equations" and I've hit a roadblock on one of the questions.
1. Consider the system of equations $$\begin{aligned} x + 2y - z &= -3 \\\ \end{aligned}$$ $$\begin{aligned} 3x + 5y + kz &= -4 \\\ \end{aligned}$$ $$\begin{aligned} 9x + (k+13)y + 6z &= 9 \\\ \end{aligned}$$ a) Express these equations as an augmented matrix
which I think is (correct me if I'm wrong): $$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 3 & 5 & k & -4 \\ 9 & (k+13) & 6 & 9 \end{array}\right] $$
I am stuck on part (b) which is:
b) Show that this matrix can be row-reduced to
$$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 0 & 0 & k^2-2k & 5k+11 \end{array}\right] $$
Guide:
Perform $R_2-3R_1$, $R_3-9R_1$, $-R_2$, and you should be one step away from the solution.