Thread: Characteristic polynomial

Hi,

Can anyone tell me the easiest way to find the characteristic polynomial of a 3x3 matrix.

I don't understand why sometimes if you can reduce it to the identity matrix -the characteristic polynomial will just be (X-1)^3 and sometimes it won't . . . (thereby forcing you to use a different method to find the characteristic polynomial)

Many thanks.

Originally Posted by jedoob

Hi,

Can anyone tell me the easiest way to find the characteristic polynomial of a 3x3 matrix.

I don't understand why sometimes if you can reduce it to the identity matrix -the characteristic polynomial will just be (X-1)^3 and sometimes it won't . . . (thereby forcing you to use a different method to find the characteristic polynomial)

Many thanks.

(X-1)^3 (eigenvectors of 1);

In order to find eigenvalues, you will become well acquainted with:

Ax = Lx (where L is lambda);

Similarly, (A - LI)x = 0, where I is the identity matrix, and x is the corresponding eigenvector.

The characteristic polynomial is simply the part before the equality (= 0). The characteristic equation is then the whole thing. What you're interested in finding is the determinant of (A - LI)x. Do you have a specific problem that you're stuck on?

Thanks. I understand this, but what I don't understand is as follows:

I have two different 3x3 matrices infront of me, they both reduce to become the identity matrix . . .

BUT for some reason-the characteristic polynomial for one of them is (X-1)^3as expected but I know that this isn't true for the other one.

Please could you explain why this should be so and also how I could find: det(LI-A) for a 3x3 matrix A directly, instead of reducing it to rref and then finding the determinent.
[NOTE: I would also need to find the minimum polynomial afterwards so pref I need a method which will give me the det factorised as much as possible.]

Well, you do not need to reduce anything in order to compute the characteristic polynomial. This is the determinant det(M-k*Id) by definition, so whatever method you like to compute a 3 x 3 determinant will do.

Originally Posted by jedoob

Thanks. I understand this, but what I don't understand is as follows:

I have two different 3x3 matrices infront of me, they both reduce to become the identity matrix . . .

BUT for some reason-the characteristic polynomial for one of them is (X-1)^3as expected but I know that this isn't true for the other one.

Please could you explain why this should be so and also how I could find: det(LI-A) for a 3x3 matrix A directly, instead of reducing it to rref and then finding the determinent.
[NOTE: I would also need to find the minimum polynomial afterwards so pref I need a method which will give me the det factorised as much as possible.]

I'm curious. Could you post these examples?

-Dan