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existence of matrices

Source: LIMIT 2019 CCS1 P4

April 28, 2021
linear algebramatrix

Problem Statement

Which of the following are true? <spanclass=′latex−bold′>(A)</span> ∃A∈M3(R) such that A2=−I3<span class='latex-bold'>(A)</span>~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3 <spanclass=′latex−bold′>(B)</span> ∃A,B∈M3(R) such that AB−BA=I3<span class='latex-bold'>(B)</span>~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3 <spanclass=′latex−bold′>(C)</span> ∀A∈M4,det⁡(I4+A2)≥0<span class='latex-bold'>(C)</span>~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0 <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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