Solving homogeneous system involving decimal eigenvalues

[maxfails]

Homework Statement

I need to find the general solution of the system
[3 5]
[-1 -2]

Homework Equations

so to get the eigenvalues, det(A - λI)

The Attempt at a Solution

determinant is (3-λ)(-2-λ) + 5

which would be λ² - λ - 1

so by the quadratic equation the eigenvalues are λ = 1/2 + (√5)/2 and 1/2 - (√5)/2

but now I don't know how to reduce the matrix to get the eigenvectors?

I think the matrix for the first eigenvalue would be
[3-(1/2 + √5)/2) 5]
[-1 -2-1/2+(√5)/2]

What to do now?

 

[Mark44]

Homework Statement

I need to find the general solution of the system
[3 5]
[-1 -2]

Homework Equations

so to get the eigenvalues, det(A - λI)

The Attempt at a Solution

determinant is (3-λ)(-2-λ) + 5

which would be λ² - λ - 1

so by the quadratic equation the eigenvalues are λ = 1/2 + (√5)/2 and 1/2 - (√5)/2

but now I don't know how to reduce the matrix to get the eigenvectors?

I think the matrix for the first eigenvalue would be
[3-(1/2 + √5)/2) 5]
[-1 -2-1/2+(√5)/2]

What to do now?

Do what you would normally do - row reduce the matrix, but you'll need to fix some misplaced parentheses first.

The matrix should be
[3- (1/2 + √5/2) 5]
[-1 -2- (1/2+√5/2)]

=

[5/2 - √5/2) 5]
[-1 -5/2 - √5/2]

You had a few too many parentheses in your matrix.

When the matrix is row-reduced, there should be a row of zeroes, and you can use the other row to get your eigenvector.