I've been stumped on this question for the past few days.
The question asks that the following augmented matrix be row-reduced to a 'goal' matrix:
\begin{matrix} 1 & 2 & -1 &| -3\\ 3 & 5 & k &| -4\\ 9 & (k+13) & 6 &|+9\\ \end{matrix}
Needs to be reduced to:
\begin{matrix} 1 & 2 & -1 &| -3\\ 0 & 1 & -k-3 &| -5\\ 0 & 0 & k^2-2k &|5k+11\\ \end{matrix}
I must have tried this upwards of 15 times - I can get the three zeroes just fine usually, but the unknowns (k) are rarely anywhere near the 'goal' matrix. I have tried getting the three zeroes in different orders, but that doesn't seem to help either.
The closest I've come is
\begin{matrix} 1 & 2 & -1 &| -3\\ 0 & 1 & -k+3 &| +5\\ 0 & 0 & -k^2-8k+12 &|-5k+7\\ \end{matrix}
I feel like I'm really close, but I just can't get the correct unknowns...
Any help would be greatly appreciated.
Thanks,
John