# For what values of a does the following homogeneous system have a nontrivial solution

1. Oct 15, 2009

### nietzsche

1. The problem statement, all variables and given/known data

For what values of a does the system

(a-1)x + 2y = 0
2x + (a-1)y = 0

have a nontrivial solution?

2. Relevant equations

3. The attempt at a solution

Argh... I'm really bad at linear algebra, can't seem to grasp the concepts.

I know that if I make this into a matrix

$$\left[ \begin{array}{cc} a-1 & 2\\ 2 & a-1 \end{array} \right]$$

that the only way the system will have a nontrivial solution is if the reduced row echelon form is not the 2x2 identity, i.e. the matrix is singular. But I don't know how to use those ideas to solve it. Any help would be great!

2. Oct 15, 2009

### Pengwuino

Re: For what values of a does the following homogeneous system have a nontrivial solu

If the determinant is 0, what does that tell you about the system?

3. Oct 15, 2009

### nietzsche

Re: For what values of a does the following homogeneous system have a nontrivial solu

For some reason, my prof hasn't taught us anything about the determinant. I've read about it, and I sort of have an idea of how it works, but I don't think our prof wants us to use it to solve problems.

4. Oct 15, 2009

### Dick

Re: For what values of a does the following homogeneous system have a nontrivial solu

Ok, so start reducing it to echelon form. Divide the first row by a-1. Under what conditions can you fail to get ones on the diagonal?

5. Oct 15, 2009

### nietzsche

Re: For what values of a does the following homogeneous system have a nontrivial solu

my intuition is telling me that a = 3 ... i did it like this:

$$\left[ \begin{array}{cc} 2 & a-1\\ a-1 & 2 \end{array} \right] \\ \text{then} \\ \left[ \begin{array}{cc} 1 & \frac{a-1}{2}\\ \frac{a-1}{2} & 1 \end{array} \right]$$

so from this we see that $$\frac{a-1}{2} = 1$$ because that would make row 2 a multiple of row 1, and subtracting it out would leave a row of zeros. so a = 3.

does it make sense? i feel like i'm missing something.

6. Oct 15, 2009

### Dick

Re: For what values of a does the following homogeneous system have a nontrivial solu

That makes a lot of sense. Sure a=3 is a problem. But keep going with the row reduction. Multiply the first row by (a-1)/2 and subtract it from the second row. There is another value of a that creates a problem.

Last edited: Oct 15, 2009
7. Oct 15, 2009

### nietzsche

Re: For what values of a does the following homogeneous system have a nontrivial solu

i was thinking that maybe a = 1 is also a problem, but if you substitute a = 1 into the original matrix, you can transform it into the 2x2 identity.

but if i try to put it into rref i end up having to divide by a-1, and if a=1 then it is undefined.

8. Oct 15, 2009

### Dick

Re: For what values of a does the following homogeneous system have a nontrivial solu

Like I edited my last post to say, just multiply the first row by (a-1)/2 and subtract from the second row. Then ask yourself when you can get a second row of zeros.

9. Oct 15, 2009

### nietzsche

Re: For what values of a does the following homogeneous system have a nontrivial solu

ah, i see

a = -1

thanks again dick. you're really saving my skin.

10. Oct 15, 2009

### Dick

Re: For what values of a does the following homogeneous system have a nontrivial solu

Well, you are helping by using the hints and thinking about them. Not everyone does that.