What is det(-B), If A is a 4x4 matrix with det(A)=3 and B can be obtained with the following row reduction operations

Question

So the row operations I was given are :

1) E2 => -2E2

2)E1 <-> E3

3)E4=>E2-2E3

4)E1=>E1+3E2

So i did calculate the determinant of matrix B as 6

(-2)x(-1)x(1)x(1)x(3)=6

But i really cant find any resource on how det(B) relates to det(-B). That's where I am having issues. I know that [-B] is the negation of all the elements in [B], but I am having a hard time joining that with the concept of determinants. I couldn't find any examples online about this either. Nor does my textbook have anything that seems helpful.

Answer 1

If $B$ is a $n\times n$ matrix then $\rm{det} (-B)=(-1)^n\rm{det} (B)$, you can see this using the definition of determinant that takes n elements of the matrix, multiplies them and sum them following the permutation form, if all of them change by -1 then all the elements of the sum will change by (-1)^n factor.