MathDB
multi-choice matrix statements

Source: LIMIT 2019 CCS1 P1

April 28, 2021
linear algebramatrix

Problem Statement

Which of the following are true? <spanclass=′latex−bold′>(A)</span> ∀A∈Mn(R),At=X−1AX for some X∈Mn(R)<span class='latex-bold'>(A)</span>~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R) <spanclass=′latex−bold′>(B)</span> ∀A∈Mn(R),I+AAt is invertible<span class='latex-bold'>(B)</span>~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible} <spanclass=′latex−bold′>(C)</span> tr⁡(AB)=tr⁡(BA),∀A,B∈Mn(R) but ∃A,B,C such that tr⁡(ABC)≠tr⁡(BAC)<span class='latex-bold'>(C)</span>~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC) <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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