cmi-entrance 2025 Q1

cmi-entrance · India · pgmath 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning
Let $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ be a linear transformation such that $T \neq 0$ and $T ^ { 4 } = 0$. Pick the correct statement(s) from below.
(A) $T ^ { 3 } = 0$.
(B) $\operatorname { Image } ( T ) \neq \operatorname { Image } \left( T ^ { 2 } \right)$.
(C) $\operatorname { rank } \left( T ^ { 2 } \right) \leq 1$.
(D) $\operatorname { rank } ( T ) = 2$.
Let $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ be a linear transformation such that $T \neq 0$ and $T ^ { 4 } = 0$. Pick the correct statement(s) from below.\\
(A) $T ^ { 3 } = 0$.\\
(B) $\operatorname { Image } ( T ) \neq \operatorname { Image } \left( T ^ { 2 } \right)$.\\
(C) $\operatorname { rank } \left( T ^ { 2 } \right) \leq 1$.\\
(D) $\operatorname { rank } ( T ) = 2$.