Question

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4$

$\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|$is $8.$

$\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|=8$

Step-by-step solution

Step 1. Given information 

The given determinant is

$\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|$

Step 2. Value of determinant 

Use the property of determinant and take $2$common form last row of determinants

$$\begin{gathered} \left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|=\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2\left(1\right)& 2\left(2\right)& 2\left(3\right)\end{array}\right| \\ =2\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right| \\ =2\times 4 \\ =8 \end{gathered}$$

So the value of the determinant role="math" localid="1647384410681" $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 2& 4& 6\end{array}\right|$is$8$.