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We know that if the states that we put in the Slater's determinant is linearly dependent, then it vanishes.

However, is the reverse true in the sense that, if the states are linearly independent, then the Slater's determinant does not vanish?

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Yes this is true. The determinant of a matrix an $m\times m$ matrix is zero if and only if the rank of that matrix is less then $m$ i.e. if the one of the columns is linearly dependent see this MSE post.

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