Q. 47 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that 

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ 2u& 2v& 2w\end{array}\right| \end{gathered}$$

Step-by-Step Solution

Answer

The value of the determinant is $8$. 

Step 1. Given Information

We are given, 

$$\begin{gathered} \left|\begin{array}{lll}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ 2u& 2v& 2w\end{array}\right| \end{gathered}$$

We have to solve the given determinant using properties of determinants. 

Step 2. Solving the determinant

Factoring out the GCF $2$, we get

$\left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ 2u& 2v& 2w\end{array}\right|=2\left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ u& v& w\end{array}\right|$

Performing the row operation $R_2=R_2+3R_1$,

$2\left|\begin{array}{ccc}1& 2& 3\\ x-3& y-6& z-9\\ u& v& w\end{array}\right|=2\left|\begin{array}{ccc}1& 2& 3\\ x& y& z\\ u& v& w\end{array}\right|$

Interchanging the rows $R_1\text{ and }{R}_2$, we get

$=-2\left|\begin{array}{lll}x& y& z\\ 1& 2& 3\\ u& v& w\end{array}\right|$

Interchanging the rows $R_2\text{ and }{R}_3$, we get

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