Please help me with this math problem?

Question

Find a fundamental matrix for x'=(-3/4, 1/8, 1/2, -3/4)x. (2x2 matrix, -3/4 and 1/8 on the left, 1/2 and -3/4 on the right. The eigenvalues are -1 and -1/2. I got -7/4a+1/2b=0, 1/8a-7/4b=0 for the first root, but how do I solve for a and b?)

 

Answer 1

Your equations for a and b for the first root aren't right. They should be

 

1/4a + 1/2b =0

1/8a + 1/4b =0

 

As always, the two equations are multiples of each other (the second equation is 1/2 the first). You can choose a or b freely and see what the other has to be. If you let b=1, then a=-2, so (-2,1) is an eigenvector for the eigenvalue -1.

 

Similarly, an eigenvector for the eigenvalue -1/2 is (2,1).

 

The rest is as in your previous problem. :)