Question

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

role="math" localid="1647383304224" $\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|$

Short answer

The value of $\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|$is $-4$

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

Step-by-step solution

Step 1. Given information

The given determinant is

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

Step 2. Solution

On comparing both determinants, it seems that the determinant $\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|$is obtained by interchanging the first and third row of determinant $\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|$

Property of determinant state that the value of a determinant changes sign if any two rows are interchanged.

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

so the value of determinant $\left|\begin{array}{ccc}1& 2& 3\\ u& v& w\\ x& y& z\end{array}\right|=-4$