Question. If $A$ is a $4 \times 6$ matrix with rank $3$, then the reduced row echelon form of A has
(1) at least one zero row.
(2) at least one zero column.
(3) exactly two zero columns.
(4) at most three zero columns.
Since the rank is $3$, one row vector would be dependent, hence that would make up the row with zeroes, so (1) is correct. I am a bit confused as to how to relate rank of a matrix with its columns. I think the answer should be (4) at most three zero columns since the rank = $3$ and the number of columns in a 4x6 matrix are 6. Any insight?
$\endgroup$ 81 Answer
$\begingroup$Assuming that the martix is in its row reduced echelon form ,since the row rank and column rank are equal you have one $0$ row and at most three $0$ columns. Thus the options $(1)$ and $(4)$ are correct.
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