I'm currently learning about finding determinant using row operations. This method requires the values below the main diagonal to all be zero. I'm looking at this example and I don't understand the last matrix. They have done r4 = 1/2r3 + r4. However, what if I had done r4 = 2r4 + r3 ? I would still get my desired zeros below the diagonals but my last value on my diagonal becomes -13 instead of -13/2. This changes my determinant result. Why is this happening? Can I not do r4 = 2r4 + r3?
There is a result that performing
has no effect on the determinant.
This is different from performing
$$r_j = r_i + kr_j$$
If you perform
$$r_3 = r_3+ 2r_4$$
you do not change the determinant.
If you perform $$r_4 = 2r_4+r_3$$
What you are doing is actually first multiplying the $4$-th row by $2$ and then $r_4 = r_4+r_3$. Hence that is why your answer differs by a multiplication factor of $2$ from the correct answer.