Question

how would you use the the following matrix
l x y z l
l u v w l = 4
l 1 2 3 l

what would the determinant of the bottom matrix
l x y z l
l -4 -8 -12 l
l u v w l

Answer 1

Whenever we swap two rows of a matrix, the sign of the determinant changes:
If
\[\det \left[\begin{matrix}a & b \\ c & d\end{matrix}\right]= e\]
Then
\[\det \left[\begin{matrix}c & d \\ a & b\end{matrix}\right]= -e\]
Also, when we do a scalar multiplication of a row, whatever the value of that scalar multiplications is also done on the determinant.
\[\det \left[\begin{matrix}a & b \\ c & d\end{matrix}\right]= e\]
Then
\[\det \left[\begin{matrix}a & b \\ 2c & 2d\end{matrix}\right]= 2e\]

Answer 2

i multiplied the it just by -4 didnt know the sign changes! Thanks alot!