Finding a determinant using row reduciton and co-factor expansion

Question

| 2 5 -3 -1 | row1

| 3 0 1 -3 | row2

|-6 0 -4 9 | row3

|4 10 -4 -1 | row4

I'm trying to get the determinant of this matrix. So my plan was to get the second column of the matrix to all zeros except for the first entry. Thus, I did

row1*2 - row 4 = |0 0 -2 -1| and replaced the last row with this row. Using the second column, I did co-factor expansion and managed to get

|3 1 -3|

|-6-4 9| * -5

|0 -2 -1|

Then I used the rule of sarrus to get the determinant of the 3x3 matrix which was 24.

So 24*-5 = -120.

But the answer was +120. Where did I miss to put a negative sign in my calculations?

Answer 1

You should have done the substitution $\;R_4\to R_4-2R_1\;$ , because if you did what you wrote there $\;R_4\to -R_4+2R_1\;$ then you multiplied your fourth row by $\;-1\;$ and then you added twice the first row: your determinant gets multipplied by $\;-1\;$