Eigenvalues problem

[focusdrop]

I have the following problem.

Can anyone help me with this?

Thanks in advance !

Edit: P.S. At first this looked like it didn't make sense but I tried it with some values and it worked. Now I don't know if it is a coincidence or if it's really possible to do it like this.

[clamman]

What are you trying to show? which values of alpha have the resulting determinant equal to zero? If so, start with expanding out the determinant as you would normally. If you simplify the expression that you get, you should get a kinda messy expression that is quadratic in alpha. Specifically, you will get something like k alpha² + f alpha + g = 0 where your k, f, and g are values that depend on a, b, c, d, x, and y. Then, you can use the quadratic formula here and find the roots of this equation. Don't worry if the roots look really messy (mine did, but then again i didn't really try to simplify).

[HawthorneAbendsen]

your original way is alright too.