Show that if A is an elementary matrix, then at least one entry in the third row must be zero?

Question

Show that if

A = ( 1 0 0 )
( 0 1 0 )
( a b c )

Show that if A is an elementary matrix, then at least one entry in the third row must be zero

Answer 1

An elementary matrix is one formed from I by applying one of the elementary row operations. Consider each ERO:

Scalar multiplication: (a b c) must be a multiple of (0 0 1), so two entries must be 0.

Interchange of rows: not possible since no other rows have been changed, but in any case would have two entries 0.

Add a multiple of R1 or R2 to R3: we either add (k 0 0) or (0 k 0) to (0 0 1), for some nonzero k. We will obtain either (k 0 1) or (0 k 1), so in either case one entry will be 0.

So at least one entry must be 0.