Given the system of equations:
$x_1 + 5x_3 = 1$
$2x_1 – x_2 + 7x_3 = 7 $
$2x_2 + cx_3 = – 1 $
where $c$ is a constant.
(the number behind the $x$ is supposed to be a number lower than the $x$, I just don't know how to use the system in here to write equations yet).
(a) Write down the extended matrix. Find the solution using Gaussian elimination method. For what values of $c$ and equation unique solution? Interpret the meaning of other values for $c$ (e.g. forbidden values).
(b) Assume that the system of equations above can be written in matrix form $Ax = B$. Let $c = 3$ Set up $A, x, B$. Calculate the determinant of $A$.
(c) Compute the inverse of $A$ by means of the adjoint matrix (matrix of cofactors). Use the result for $A -1$ to find the solution to equation set.
I'm not very good at math but I've tried this one. My books is really bad at explaining things and I don't really know anyone who is good at math. I would very much appreciate help with this problem.
I've read up on homework questions in here and also sent my professor an email, he said it was fine as long as I afterwards were able to show that I understood it, not just copy pasted of someone else.
(b) and (c) I'm really in the dark about these two, but I think I've understood somewhat how to do (a).
The extended matrix:
$\begin{bmatrix} 1 & 0 & 5 & 1 \\ 2 & -1 & 7 & 7 \\ 0 & 2 & c & -1 \\ \end{bmatrix} $
And in (a) I multiplied the first line and mixed it with the second to remove ($2x_1$), and in the third line (where my problem is) I got $(c-6)x_3 = 9$.
So, I'd love some help with $a-c$, but if that's not allowed It would be very helpful if someone could explain what to do with the $c$ in the third line. What I understood is that if I manage to get $x_3 = "something"$ I could use that in the other lines to find $x_1$ and $x_2$ and that would mean that I've solved it using Gaussian elimination right?