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I couldn't give the full explanation in the title - I am talking about a particular matrix. Given the matrix:

A[1] =

0 0 1

1 0 0

1 1 0

A[5] =

1 1 1

1 1 1

1 1 1

Once it gets to the 5th boolean power, it becomes all 1's, and any power greater than or equal to 5 will always produce a matrix of all 1's. I know that this will always be true because A[5], A[6], A[N] will always have only 1's, and that since A[1] has at least one "1" in each column, that will always be enough to produce a one in each column, when taking the boolean product of each row of A. I understand this, but am not sure how to put it into a proof. Would appreciate any tips.

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# Prove that a zero-one matrix can only have 1's after 5th power

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