cmi-entrance 2024 Q8

cmi-entrance · India · mscds Matrices True/False or Multiple-Select Conceptual Reasoning
8. Which of the following statements are valid for all $n \times n$ matrices $A , B$ :
(a) $\left( A ^ { T } A \right) ^ { T } = A A ^ { T }$.
(b) If $A , B$ are invertible, then inverse of $A B$ is $A ^ { - 1 } B ^ { - 1 }$.
(c) $( A + B ) ^ { T } = A ^ { T } + B ^ { T }$
(d) $A x = B x$ for some $n \times 1$ vector $x$ implies that $A y = B y$ for all $n \times 1$ vectors $y$.
8. Which of the following statements are valid for all $n \times n$ matrices $A , B$ :\\
(a) $\left( A ^ { T } A \right) ^ { T } = A A ^ { T }$.\\
(b) If $A , B$ are invertible, then inverse of $A B$ is $A ^ { - 1 } B ^ { - 1 }$.\\
(c) $( A + B ) ^ { T } = A ^ { T } + B ^ { T }$\\
(d) $A x = B x$ for some $n \times 1$ vector $x$ implies that $A y = B y$ for all $n \times 1$ vectors $y$.\\