# Find the eigenvalues of a given matrix

1. Apr 14, 2013

### blouqu6

1. The 3x3 Matrix A=[33, -12, -70; 0, 1, 0; 14, -6, -30] has three distinct eigenvalues, λ123.
Find each eigenvalue.

2. det(A-λI)=0 where I denotes the appropriate identity matrix (3x3 in this case)

3. Here's my attempt:

--> det([33, -12, -70; 0, 1, 0; 14, -6, -30]-λ[1, 0, 0; 0, 1, 0; 0, 0, 1])=0

--> det([33-λ, -12, -70; 0, 1-λ, 0; 14, -6, -30-λ]=0

--> -λ3+4λ2+7λ-10=0

And this is where I'm not exactly sure what to do. I don't believe I can effectively use grouping to solve for λ. Any help here would be appreciated.

Last edited: Apr 14, 2013
2. Apr 14, 2013

### Dick

The rational roots theorem. Any possible rational root must divide 10. Can you guess one? Once you find a root r, divide by λ-r. Now you have a quadratic.

3. Apr 14, 2013

### blouqu6

Ahh yes

And that did it. Thanks man, I ended up with λ1=-2,λ2=1, and λ3=5, which is the correct answer. Really appreciate the help.

4. Apr 14, 2013

### Dick

You are welcome. If they give you a cubic to solve without using a computing device, it will likely have one easy root. If you find that you are home free.