- #1

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## Homework Statement

3.For which values of ##\lambda## does the following system of equations also have non trivial solutions

## Homework Equations

## The Attempt at a Solution

What I tried doing first is to put all variables on the same side and got

##

v+y-\lambda*x=0\\

x+z-\lambda*y=0\\

y+u-\lambda*z=0\\

z+v-\lambda*u=0\\

u+x-\lambda*v=0

##

and when I wrote the coefficient into the matrix i got

##

\begin{bmatrix}

-\lambda& 1 &0&0&1\\

1&-\lambda&1&0&0\\

0&1&-\lambda&1&0\\

0&0&1&-\lambda&1\\

1&0&0&1&-\lambda\\

\end{bmatrix}

##

here I noticed that all the columns sum to the same number ##2-\lambda## there I summed everything into the first row and got

##

\begin{bmatrix}

2-\lambda & 2-\lambda&2-\lambda&2-\lambda&2-\lambda\\

1&-\lambda&1&0&0\\

0&1&-\lambda&1&0\\

0&0&1&-\lambda&1\\

1&0&0&1&-\lambda\\

\end{bmatrix}

##

here I looked into 2 different possibilities if a) ##\lambda=2## and b) ##\lambda\neq2##.

However a) is pretty simple and it's mostly b) that I'm having trouble with.

Here I thought if ##\lambda\neq2## then I can devide the first row by ##2-\lambda##

When I did this my matrix looked like this

##

\begin{bmatrix}

1 & 1&1&1&1\\

1&-\lambda&1&0&0\\

0&1&-\lambda&1&0\\

0&0&1&-\lambda&1\\

1&0&0&1&-\lambda\\

\end{bmatrix}

##

Then I subtracted the first row from the second and last one and got

##

\begin{bmatrix}

1 & 1&1&1&1\\

0&-\lambda-1&0&-1&-1\\

0&1&-\lambda&1&0\\

0&0&1&-\lambda&1\\

0&-1&-1&0&-\lambda-1\\

\end{bmatrix}

##

then I just rearranged some rows so that it would be easier for me to read

##

\begin{bmatrix}

1 & 1&1&1&1\\

0&1&-\lambda&1&0\\

0&-1&-1&0&-\lambda-1\\

0&-\lambda-1&0&-1&-1\\

0&0&1&-\lambda&1\\

\end{bmatrix}

##

then I added the second row to the third and forth one and switched the third and forth row

##

\begin{bmatrix}

1 & 1&1&1&1\\

0&1&-\lambda&1&0\\

0&-\lambda&-\lambda&0&-1\\

0&0&-1-\lambda&1&-\lambda-1\\

0&0&1&-\lambda&1\\

\end{bmatrix}

##

Lastly I added the last row to the forth one and switched them

##

\begin{bmatrix}

1 & 1&1&1&1\\

0&1&-\lambda&1&0\\

0&-\lambda&-\lambda&0&-1\\

0&0&1&-\lambda&1\\

0&0&-\lambda&1-\lambda&-\lambda\\

\end{bmatrix}

##

Here is where I get stuck. I don't know how to continue from here on out. Maybe I made a mistake somewhere in my addition however I went through it at least a few times and I was not able to find it:

Any help / tips are greatly appreciated

Thanks