It's two row swaps, and a row multiplication by -1, so your answer looks right to me.
Ok the problem is,
Given:
|r s t|
|u v w| = 4
|x y z|
Evaluate:
|-x -y -z|
|r s t|
|u v w|
My steps at finding the solution are:
1.
|r s t|
(-)|-x -y -z|
|u v w|
2.
|r s t|
(-)(-)|u v w|
|-x -y -z|
3.
|r s t|
(-)(-)(-1)|u v w|
|x y z|
4. (-)(-)(-1)(4)= -4
Which for some reason is wrong because I am supposed to show that it equals 4 not -4. What am I doing wrong?
How did you get to from the original matrix ?
First, you interchanged the 2nd and 3rd rows, and thus the determinant is multiplied by -1, and then you interchanged the 1st and 2nd rows and again the det. gets multiplied by -1, so we still have the original determinant. But now you multiplied the first row by -1 and thus the det. of the new matrix equals ...
Thus you're right: it is -4
Tonio