# Gaussian Elimination and Matrix question

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?

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I believe this question should have a label "homework" not to confuse the readers. "and in the third line (where my problem is) I got (c−6)x3=9" It is not a problem, you just have to express all the x_i variables as a function of c. To be more specific once you have x_3 as you do, you do the upwards sweeping portion of the Gaussian elimination i.e. eliminate the remaining off diagonal terms line by line. As a result all the x_i variables should be expressed as functions of c. About all the other parts you should consult your lecture notes / Wikipedia on how to compute determinants etc. they ar – peterm Oct 8 '12 at 15:49

## 1 Answer

Your system of equation give the following matrix as you did: $$\begin{bmatrix} 1 & 0 & 5 & |1 \\ 2 & -1 & 7 & |7 \\ 0 & 2 & c & | -1 \\ \end{bmatrix}$$ Now by doing $-2R_1+R_2\rightarrow R_2$ you have $$\begin{bmatrix} 1 & 0 & 5 & |1 \\ 0 & -1 & -3 &| 5 \\ 0 & 2 & c & |-1 \\ \end{bmatrix}$$ and by $2R_2+R_3\rightarrow R_3$ you have $$\begin{bmatrix} 1 & 0 & 5 & |1 \\ 0 & -1 & -3 &| 5 \\ 0 & 0 & c-6 & |9 \\ \end{bmatrix}$$ Now you can arrange a new system of equations by using the final matrix exactly as you get the first extended matrix. I mean: $$x_1+5x_3=1\\-x_2-3x_3=5\\(c-6)x_3=9$$ Suppose that $c=6$ then what happens? A contradiction! In this case the system has no solution as form $(x_1,x_2,x_3)$. On the other hand you can read the determinant from the final matrix. It is $1*(-1)*(c-6)=6-c$. If you want your non homogenous system has a unique solution; the determinant should not be zero. It means that $c$ must be any number than $6$.

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Opss! I made a mistake. Sorry. Fixed it. $-x_3$ should have been $-3x_3$ in the second equation.:) – Babak S. Oct 8 '12 at 19:16
@FrankLen: I used the elementary row operations to reduce a matrix to what is called triangular form (the last matrix). Always, when you want to find the determinant of a square matrix $A$; you can do these processes and you did it right for this system. When the matrix becomes upper triangular; you can multiply the diagonal elements and find it 6-c. – Babak S. Oct 8 '12 at 19:27
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