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?
- Any row echelon form of A has 3 non-pivot columns.
- Any row echelon form of A has 2 zero rows.
- A is not full rank.
- 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.
Best Answer
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.