GATE 2016 | MATHS | Q-14

Question

Consider a real vector space \( V \) of dimension \( n \) and a non-zero linear transformation \( T: \mathbb{V} \rightarrow \mathbb{V} \). If \( \text{dim}(T) < n \) and $T^2 = \lambda T$, for some \( \lambda \in \mathbb{R} \backslash \{0\} \), then which of the following statements is TRUE?
(A) \( \text{det}(T) = |\lambda|^{2} \)
(B) There exists a non-trivial subspace \( \mathbb{W} \) of \( V \) such that \( T(X) = 0 \) for all \( X \in \mathbb{W} \)
(C) \( T \) is invertible
(D) \( \lambda\) is the only eigenvalue of \( T \)