Theorem: If A is a square matrix whose column sums are all equal to 1, then the equation AX=X has non - trivial solution.
Proof:
Suppose, by contrapositivity, that AX=X does not have non-trivial solution. This implies that the solution is trivial. Indeed, AX=X IFF X = 0. From here, it suffices to show that the columns of A does not sum to 1.
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