I have a 3x3 matrix as follows: \begin{bmatrix}4&3&1\\2&0&2\\2&1&0\end{bmatrix} I know that the correct determinant is 6, and can obtain this value using various elementary row operation routes or other methods. However, I am confused as to why the following series of operations fails to end with a determinant of 6:
Replacing row 2 with row 1 - 2*(row 2): \begin{bmatrix}4&3&1\\0&3&-3\\2&1&0\end{bmatrix}
Replacing row 3 with row 1 - 2*(row 3): \begin{bmatrix}4&3&1\\0&3&-3\\0&1&1\end{bmatrix}
Replacing row 3 with row 2 - 3*(row 3): \begin{bmatrix}4&3&1\\0&3&-3\\0&0&-6\end{bmatrix}
This gives a determinant of -72, which is way off from the correct answer of 6. As far as I can tell, I didn't use scaling at any point, nor switched any rows. I feel as though I'm missing some kind of important restriction when using elementary row operations for this purpose, or maybe just overlooking a simple detail. Any help is appreciated.
Edit: If row reduction in the manner described above changes the determinant, why does the following sequence result in 6?
Replacing row 2 with -1/2 *(row 1) + row 2
Replacing row 3 with -1/2 *(row 1) + row 3
Replacing row 3 with -1/3 *(row 2) + row 3
Would this cause the determinant to be multiplied by -1/12?
Or is it specifically related to the multiplication of the row being replaced?
Edit 2:
Thank you, I misread the answer. I'll accept it as soon as I'm allowed.
Replacing row i with row i + n*row j doesn't change the determinant. Replacing row i with row j + n*row i does (it multiplies the determinant by n).