Let T be a linear operator on ℝ3. Let f(X) ∈ ℝ[X] denote its characteristic polynomial. Consider the following statements.
(a). Suppose T is non-zero and 0 is an eigen value of T. If we write f(X) = X g(X) in ℝ[X], then the linear operator g(T) is zero.
(b). Suppose 0 is an eigenvalue of T with at least two linearly independent eigen vectors. If we write f(X) = X g(X) in ℝ[X], then the linear operator g(T) is zero.
Which of the following is true?
1
Both (a) and (b) are true.
2
Both (a) and (b) are false.
3
(a) is true and (b) is false.
4
(a) is false and (b) is true.