1
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ Treat $k$ as an arbitrary constant, then you'll be able to use elementary row operations involving terms with $k$. $\endgroup$ – user124862 Mar 6 '14 at 2:33
  • $\begingroup$ @Amzoti I tried what you wrote, but the became too convoluted after R3=R1/9.. $\endgroup$ – user133422 Mar 6 '14 at 8:05
  • $\begingroup$ @MitchKnight That's the problem I'm having though; I can never get the constant k to be as shown in the goal matrix :( $\endgroup$ – user133422 Mar 6 '14 at 8:06
0
$\begingroup$

I was struggling with this one, so I used software to compute the reduced row echelon forms (which are unique to each linear system) for each and found them to be equal to each other, thus I need that the first matrix could be reduced to the second. Now, $$\begin{pmatrix} 1&2&-1&-3\\ 3&5&k&-4\\9&(k+13)&6&6\end{pmatrix} -3R_1+R_2\to R_2 \begin{pmatrix} 1&2&-1&-3\\ 0&-1&(3+k)&5\\9&(k+13)&6&9\end{pmatrix}$$ $$9R_1 -R_3 \to R_3 \begin{pmatrix} 1&2&-1&-3\\ 0&-1&(3+k)&5\\0&(5-k)&-15&-36\end{pmatrix} $$ $$-R_2 \to R_2 \begin{pmatrix} 1&2&-1&-3\\ 0&1&(-3-k)&-5\\0&(5-k)&-15&-36\end{pmatrix}$$ $$(5-k)R_2-R_3 \to R_3 \begin{pmatrix} 1&2&-1&-3\\ 0&1&(-3-k)&-5\\0&0&(k^2-2k)&(5k+11)\end{pmatrix}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.