Difficult Matrix Equation

Hi

I'm supposed to solve this matrix equation by GaussJordan elimination and I find it very difficult to see and carry out the row operations mainly because of the z's. If someone is able to help me I would truly appreciate it.

zx1 + x2 + x3 = 1

x1 + zx2 + x3 = 1

x1 + x2 + zx3 = 1

z has a real value.

Re: Difficult Matrix Equation

Do you have to use elimination?

Can you see that x1=x2=x3?

Re: Difficult Matrix Equation

Yes I have to use the Gauss Jordan elimination method.

Re: Difficult Matrix Equation

The fact that it is "z" rather than a number is just means you have to use algebra rather than arithmetic.

$\displaystyle \begin{bmatrix}z & 1 & 1 & 1 \\ 1 & z & 1 & 1 \\ 1 & 1 & z & 1\end{bmatrix}$

The first thing I would do, to simplify the algebra, is swap the first and third rows:

$\displaystyle \begin{bmatrix}1 & 1 & z & 1 \\ 1 & z & 1 & 1 \\ z & 1 & 1 & 1 \end{bmatrix}$

Then "clear" the first column by subtracting the first row from the second and subtracting z times the first row from the third:

$\displaystyle \begin{bmatrix}1 & 1 & z & 1 \\ 0 & z- 1 & 1- z & 0 \\ 0 & 1- z & 1- z^2 & 1- z\end{bmatrix}$

Now divide the second row by z-1 to get

$\displaystyle \begin{bmatrix}0 & 1 & -1 & 0\end{bmatrix}$

and clear the second row by subtracting the second row from the first and subtracting 1- z times the second row from the third.

You try to finish it.