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Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix: $$ M:= \begin{bmatrix} \cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\ \cPr{X=...