Please help me with this math problem?

Question

Find the eigenvalues in terms of α for x'=(5/4, alpha, 3/4, 5/4)x. (this is 2x2 matrix, 5/4 and alpha on the left, 3/4 and 5/4 on the right. I got up to r^2-(5/2)r+25/16-(3/4)alpha=0, what's next?)

Answer 1

What's next is to use the quadratic formula to solve for r.

 

However, in this example, you can actually avoid having to use the quadratic formula if you go back one step and don't square out the complete square in the characteristic polynomial. You had

 

(5/4-r)²-3/4 α=0

 

Therefore,

 

(5/4-r)² = 3/4 α

 

5/4-r = ±√(3/4 α)

 

r = 5/4±√(3/4 α)

 

(This method works whenever all diagonal elements in your matrix are the same.)