Question

Is there such a thing as a matrix without pivot positions?

Answer 1

as is, yes

Answer 2

2 3
4 -1

as is, has no pivot positions, but if we are meant to relate this to its row reduced form, then maybe

Answer 3

can you think of a matrix (does it have to be nxn?) that doesnt reduce to an echelon form?

Answer 4

well 2 4
3 6 0 0
would reduce down to 3 6.
but I'm struggling with trying to show a 2x2 case generally that doesn't have leading ones but dont know where to start.. or if I even can. Ultimately, i'm trying to prove something by contradiction.

Answer 5

oops that's supposed to be 24
36

and 00
36.