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: [tex] P_{AB}(x)=P_{BA}(x) [/tex]The Attempt at a Solution
Since A is invertible we have det(A)=0. [tex]det(AB)= det(A)det(B)=0det(B)=0 ->P_{AB}(x)=P_{BA}(x)=0 [/tex]
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
A characteristic polynomial is a polynomial function that is used to find the eigenvalues of a square matrix. It is typically denoted by det(A-λI), where A is the matrix and λ is the variable.
A characteristic polynomial is used in linear algebra to find the eigenvalues of a matrix, which are important in understanding the behavior of a system. It is also used in differential equations to solve for the general solutions.
The degree of a characteristic polynomial is equal to the size of the matrix, or the number of rows/columns. For a 3x3 matrix, the characteristic polynomial will have a degree of 3.
Yes, a characteristic polynomial can have complex roots. This is because the eigenvalues of a matrix can be complex numbers, and the characteristic polynomial is used to find these eigenvalues.
The characteristic polynomial is important because it helps determine the stability and behavior of a system. The eigenvalues of a matrix found using the characteristic polynomial can indicate whether the system will converge or diverge, and how quickly it will do so.