Use the properties of determinants to find the value of the second determinant, given the value of the first.
-$$\left| \begin{array} { c c c }
x & y & z \
u & v & w \
1 & 1 & 1
\end{array} \right| = - 6 \quad \left| \begin{array} { l l l }
u & v & w \
2 & 2 & 2 \
x & y & z
\end{array} \right| = ?$$
A) 12
B) -6
C) -12
D) 6

Question

Use the properties of determinants to find the value of the second determinant, given the value of the first.
-$$\left| \begin{array} { c c c } 
x & y & z \
u & v & w \
1 & 1 & 1
\end{array} \right| = - 6 \quad \left| \begin{array} { l l l } 
u & v & w \
2 & 2 & 2 \
x & y & z
\end{array} \right| = ?$$
A) 12
B) -6
C) -12
D) 6

Quizplus · Accepted Answer

The determinant of a matrix changes sign if two rows (or columns) are swapped. In the second determinant, swapping the first and third rows will return the matrix to the form of the first determinant, but with a sign change due to the swap. Additionally, multiplying a row by a scalar multiplies the determinant by that scalar. Since the second row of the second determinant is twice each element of the third row of the first determinant, the determinant is also multiplied by 2. Therefore, the value of the second determinant is $2 \times -6 = -12$.

Use the Properties of Determinants to Find the Value of the Second