This is faulty logic. **If** the system is consistent, then it is
true that it must have infinitely many solutions when there are more
variables than equations, but it is also possible for the system to
be inconsistent.

Someone also asked about how to do the row reduction. Click here for a hint:

To avoid fractions, first perform the following row operations:

Scale row 1 by 2.

Scale row 2 by 6.

Scale row 3 by 3.

Then proceed as usual:

Add -1 times row 1 to row 2.

Add -1 times row 1 to row 3.

Add 2 times row 2 to row 3.

Add -2 times row 2 to row 1.