I'm pretty new to matrices, and only know the basics at the moment.
I was given a question that asks:
For the system of equations:
$$
\begin{split}
x &+ 2y &+ z &=60\\
2x &+ 3y &+ z &=85\\
3x &+ y &+ pz &=105
\end{split}
$$
find the value of $p$ such that there is an inconsistency and hence no solutions.
I converted the system into an augmented matrix:
$$ \left[ \begin{array}{ccc|c} 1&2&1&60\\ 2&3&1&85\\ 3&1&p&105 \end{array} \right] $$
I've tried to reduce the matrix as much as I can, but I'm pretty sure I made a mistake at some point:
$$ \left[ \begin{array}{ccc|c} 1&0&-1&-10\\ 0&1&1&35\\ 0&0&(p-8)&-100 \end{array} \right] $$
How do you find the value of $p$ that makes these equations inconsistent?
And, if I did mess up the reduction, could someone please walk me through the reduction process?
Apologies if this question has been asked before, I had a good look around and didn't find anything matching this situation.
PS: I've not learnt the determinant yet.
UPDATE
As I thought, I messed up the reduction (only slightly, though). The correct reduced form is:
$$ \left[ \begin{array}{ccc|c} 1&0&-1&-10\\ 0&1&1&35\\ 0&0&(p+2)&100 \end{array} \right] $$
(Can't be reduced any further without dividing by $p$.)
For a system to be inconsistent, it must have a row of $ \left[ \begin{array}{ccc|c} 0&0&0&n\end{array} \right] $, where $n\neq0$.
Therefore, as many of you had already deduced, $p=-2$, since then $ \left[ \begin{array}{ccc|c} 0&0&0&100\end{array} \right] $.
Thanks to everyone for helping!
Best Answer
The determinant of the coefficient for
$$\left[ \begin{array}{ccc|c} 1&2&1&60\\ 2&3&1&85\\ 3&1&p&105 \end{array} \right]$$
is found to be $$1(3p-1)-2(2p-3)-7$$
Solve $$1(3p-1)-2(2p-3)-7=0$$ and you get $$p=-2$$