Question

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

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

Short answer

The value of the determinant is $8$.

Step-by-step solution

Step 1. Given Information 

We are given,

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

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

Step 2. Solving the determinant 

Factoring out the GCF 2$2$, we get

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

Performing row operation $R_3=R_3+R_1$, we get

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

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

role="math" localid="1647417553137" $=-2\left|\begin{array}{lll}x& y& z\\ 1& 2& 3\\ u& v& w\end{array}\right|$

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

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