MathDB

Problem 4

Part of 2019 LIMIT Category C

Problems(2)

existence of matrices

Source: LIMIT 2019 CCS1 P4

4/28/2021
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}
linear algebramatrix
conditional distribution

Source: LIMIT 2019 CCS2 P4

4/28/2021
Let X,YX,Y be i.i.d Geom⁡(p)\operatorname{Geom}(p). What is the conditional distribution of X∣X+Y=kX|X+Y=k? <spanclass=′latex−bold′>(A)</span> Uniform⁡{1,2,…,⌊k2⌋}<span class='latex-bold'>(A)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\} <spanclass=′latex−bold′>(B)</span> Uniform⁡{1,2,…,k}<span class='latex-bold'>(B)</span>~\operatorname{Uniform}\left\{1,2,\ldots,k\right\} <spanclass=′latex−bold′>(C)</span> Uniform⁡{1,2,…,⌊k2⌋+1}<span class='latex-bold'>(C)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
probability