試題詳解

9. (10%) Which of the following statements are true?
(A) Let A and B be n✖ n matrices. Assume that A is invertible and B³ = 0. If AB=BA, then A+B is also invertible.
(B) Let S be the set of all sequences in R^∞ which have exactly N non-zero elements where N is a known constant. S is a subspace of  R^∞.
 
(C) Any square matrix A can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.
 
(D) If A and B are similar, they have the same eigenvectors.

(E) If A, B rank (A- B) ≤ rank( A )-rank( B )