Is it possible for a matrix to have no leading ones?

Question

Is it possible for a matrix to have no leading ones? So that its reduced row echelon form has no leading ones. Is this possible?

Would the empty set be the only possible solution to this problem?

Answer 1

The zero matrix, of any size, is in reduced row echelon form with no leading ones. As soon as there is a nonzero entry, there will be at least a leading one.

Answer 2

Not quite the empty set. The obvious answer is $A=0$.

Answer 3

If the reduced row echelon form has no leading ones, it is the zero matrix, and therefore any matrix which reduces to that is already the zero matrix.