I am having trouble understanding a problem that my Linear Algebra class gave. I understand that determinants can be found through row operations with the following points:
1.) Adding a multiple of one row to another - Doesn't change the determinant.
2.) Interchanging two rows - Reverse the sign of the determinant.
3.) Scaling a row by some constant $s$- Multiplies the determinant by that constant $s$.
Finally, we take it as noted that the determinant of the elementary square matrix is 1.
Now, I understand you can use the simple short cut of multiplying across the diagonal but I am confused about the long way of finding it (with the rules above).
You can see that to make this turn into the elementary matrix you must:
1.) Multiply row 2 by 1/2
2.) Multiply row 3 by 1/7
According to the scaling rule for finding determinants with row operations (number 3 above), I must scale the determinate by 1/2 and then by 1/7, not 2 and then 7. Why did they do it by 2 and 7? What is flawed in my understanding here?