Finding Characteristic Polynomial of Matrix B

[Boom101]

Homework Statement

B =
|a 1 -5 |
|-2 b -8 |
|2 3 c |

Find the characteristic polynomial of the following matrix.

Homework Equations

None

The Attempt at a Solution

So basically I have to find the det(B-λI). No matter what I do to the matrix I can't make the result simple. I can't find a simple answer using any cofactor expansion of the matrix. I get to (a-λ)[(b-λ)(c-λ)+24)] - [(-2c+2λ)+16] -5[-6-(2b-2λ)] which comes out to about 20 different characters. I tried rearranging the matrix to add a 0 to help with cofactor expansion but that doesn't help cause I'm left with like b+3-λ which doesn't help. Is there just no simple answer or am I doing something wrong?

 

[micromass]

Your characteristic polynomial seems correct to me. You only need to simplify it further...

 

[Boom101]

Ok thanks. I just have another question. Let M be an nxn matrix this is invertible and let A be an nxn matrix. show that the eigenvalues of (inverse of M) are the roots of the polynomial p(λ) = det(A-λM). I just have no idea where to go on this one.

 

[Mark44]

Ok thanks. I just have another question. Let M be an nxn matrix this is invertible and let A be an nxn matrix. show that the eigenvalues of (inverse of M) are the roots of the polynomial p(λ) = det(A-λM). I just have no idea where to go on this one.

What's the exact problem statement? I haven't done any work on this, but there don't seem to be enough conditions on A and M for this to be true.

However, for this to be true, if λ₁ is an eigenvalue of M^-1, then for some nonzero vector x, M^-1x = λ₁x.

 

[Boom101]

That was the exact problem statement. It was b) from which a) was the question above. Regarding my first question, I'm left with about 20 different combinations of variables which I can't simplify. If I can't simplify, should I just leave it the way it is?

 

[Mark44]

The problem is just asking for the characteristic polynomial. You could multiply things out and give it in terms of λ³, λ², λ, and a constant.

For the second question you asked, it seems to me that you're supposed to conclude that matrix A is actually the identity matrix I.

 

[Boom101]

Thanks again mark.

 

[Boom101]

Thanks again mark.