The discussion revolves around proving that for two n x n matrices A and B, where A is invertible, the characteristic polynomials satisfy P_{AB}(x) = P_{BA}(x). The initial attempt incorrectly states that since det(A) = 0, it leads to P_{AB}(x) = P_{BA}(x) = 0. However, it is established that an invertible matrix has a non-zero determinant, which is crucial for defining the characteristic polynomial correctly. The conversation emphasizes the importance of understanding the relationship between determinants and characteristic polynomials.
PREREQUISITESStudents and educators in linear algebra, particularly those focusing on matrix theory and its applications in eigenvalue problems.
Dick said:netheril96, you are supposed to give hints and help. Not work out people's problems for them. Read the forum guidelines.
talolard said: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?
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netheril96 said:But this problem is so easy that the only hint I can give is the direct answer
Dick said: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 said:Deleted