[Math] Finding determinant using row operations

determinantlinear algebra

Photograph of worked example in textbook

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?

Best Answer

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.