Discussion
Let A and B be similar matrices. Consider the following statements :
(i) tr A = tr B
(ii) det A = det B
(iii) Characteristic polynomial of A is equal to the characteristic polynomial of B
(iv) Minimal polynomial of A is equal to the minimal polynomial of B
Answer: A
All the statements are true.
If \(B = P^{-1}AP\), then for any polynomial \(g\), \(g(B) = P^{-1}g(A)P\), so \(g(A) = 0\) exactly when \(g(B) = 0\): the minimal polynomials agree. The characteristic polynomials agree because
\[ \begin{aligned} \det(B - \lambda I) &= \det(P^{-1}(A - \lambda I)P) \\ &= \det(A - \lambda I) \end{aligned} \]
The trace and the determinant are coefficients of the characteristic polynomial (up to sign), so they agree as well. Every option that leaves out one of (i) to (iv) is incomplete.
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