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Given A is a 4 x 5 matrix with rank 2, are any of the below statements not always true or are all 4 always true?

  1. Any row echelon form of A has 3 non-pivot columns.
  2. Any row echelon form of A has 2 zero rows.
  3. A is not full rank.
  4. The first two rows of A are non-zero.

Based on what I've been taught in class, 4 is not always true. For 1, rank is a number of pivot-columns of the REF of A, so if rank is 2, it the REF should have 3 non-pivot columns. For 3, full rank means rank(A) less than or equal to the smaller of its size. Since rank is 2 and not 4, A is not full rank.

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Everything that you have said is correct. 1 and 3 are always true, but 4 is not always true. You didn't say anything about 2; statement 2 is also always true.

So, 4 is the only statement that is not always true.

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