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By characteristic polynomial of A i mean det(A-tI) where t is a scalar.

A is similar to B if A = Q^-1 B Q for some invertible matrix Q. (i.e. B is the matrix representation of the same linear transformation as A but under a different basis.)

I do know that similar matrices have the same determinant. This can be easily proved using det(AB) = det(A)det(B). But when you change the diagonal entries of the determinant im not sure how it will be affected...