Do Invertible Matrices Influence the Equality of Characteristic Polynomials?

[talolard]

Homework Statement

Let A and B be to nXn matrices and A is invertible. Prove: P_{AB}(x)=P_{BA}(x)

The Attempt at a Solution

Since A is invertible we have det(A)=0. det(AB)= det(A)det(B)=0det(B)=0 ->P_{AB}(x)=P_{BA}(x)=0

Is that correct?




[

 

[Dick]

netheril96, you are supposed to give hints and help. Not work out people's problems for them. Read the forum guidelines.

 

[netheril96]

netheril96, you are supposed to give hints and help. Not work out people's problems for them. Read the forum guidelines.

But this problem is so easy that the only hint I can give is the direct answer

 

[Dick]

Homework Statement

Let A and B be to nXn matrices and A is invertible. Prove: P_{AB}(x)=P_{BA}(x)

The Attempt at a Solution

Since A is invertible we have det(A)=0. det(AB)= det(A)det(B)=0det(B)=0 ->P_{AB}(x)=P_{BA}(x)=0

Is that correct?

[

Talolard, the determinants you've written down are all just numbers. They don't have anything to do with the characteristic polynomial. What's the definition of a characteristic polynomial in terms of a determinant? Besides if A is invertible it's determinant is NOT 0.

 

[Dick]

But this problem is so easy that the only hint I can give is the direct answer

It's not 'so easy' for a lot of people. If the only hint you can think of is to give the solution, don't. Find another thread to work on. Could you remove the post with your proof in it please?

 

[netheril96]

It's not 'so easy' for a lot of people. If the only hint you can think of is to give the solution, don't. Find another thread to work on. Could you remove the post with your proof in it please?

Deleted

 

[Dick]

Deleted

Thanks!

 

[VeeEight]

You may wish to consider similar matrices here