Question

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

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

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

Short answer

The value of the determinant is $4$.

Step-by-step solution

Step 1. Given Information

We are given,

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

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

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

Step 2. Solving the determinant

Using subtraction of determinant, we get

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

Since the two rows of second determinant is same, so

$\left|\begin{array}{ccc}1& 2& 3\\ x-u& y-v& z-w\\ u& v& w\end{array}\right|=\left|\begin{array}{ccc}1& 2& 3\\ x& y& z\\ u& v& w\end{array}\right|-0$

Interchanging the rows, we get

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